第3章関数Functions

Answer Key解答

Solution 1解答 1

ⓐyesⓑyes (Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)

ⓐはいⓑはい(注意: たとえば二人の選手が4位で並んでいたら、名前は順位の関数ではなくなる)

Solution 2解答 2

$w = f (d)$

Solution 3解答 3

yes

はい

Solution 4解答 4

$g (5) = 1$

Solution 5解答 5

$m = 8$

Solution 6解答 6

$y = f (x) = \frac{\sqrt[3]{x}}{2}$

Solution 7解答 7

$g (1) = 8$

Solution 8解答 8

$x = 0$ or $x = 2$

$x = 0$ または $x = 2$

Solution 9解答 9

ⓐyes, because each bank account has a single balance at any given time;ⓑno, because several bank account numbers may have the same balance;ⓒno, because the same output may correspond to more than one input.

ⓐはい。どの銀行口座も、ある時点での残高は一つだけだから。ⓑいいえ。いくつもの口座番号が同じ残高を持ちうるから。ⓒいいえ。同じ出力が二つ以上の入力に対応しうるから。

Solution 10解答 10

ⓐYes, letter grade is a function of percent grade;ⓑNo, it is not one-to-one. There are 100 different percent numbers we could get but only about five possible letter grades, so there cannot be only one percent number that corresponds to each letter grade.

ⓐはい。文字の成績は百分率の成績の関数である。ⓑいいえ、一対一ではない。百分率の数は100通りありうるのに、文字の成績はおよそ五通りしかないので、各文字の成績に対応する百分率の数が一つだけということはありえない。

Solution 11解答 11

yes

はい

Solution 12解答 12

No, because it does not pass the horizontal line test.

いいえ。横線の判定法を通らないから。

Solution 1解答 1

${- 5, 0, 5, 10, 15}$

Solution 2解答 2

$(- \infty, \infty)$

Solution 3解答 3

$(- \infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty)$

Solution 4解答 4

$[- \frac{5}{2}, \infty)$

Solution 5解答 5

ⓐvalues that are less than or equal to –2, or values that are greater than or equal to –1 and less than 3ⓑ ${x | x \leq - 2 or - 1 \leq x < 3}$ ⓒ $(- \infty, - 2] \cup [- 1, 3)$

ⓐ–2以下の値、または–1以上3未満の値

Solution 6解答 6

domain =[1950,2002] range = [47,000,000,89,000,000]

定義域 =[1950,2002] 値域 = [47,000,000,89,000,000]

Solution 7解答 7

domain: $(- \infty, 2];$ range: $(- \infty, 0]$

定義域: $(- \infty, 2];$ 値域: $(- \infty, 0]$

Solution 8解答 8

This graph shows a piecewise function with a curve extending left, a horizontal line segment, and a line segment extending right. Open and closed circles mark function behavior at specific points, indicating discontinuities.

Solution 1解答 1

$\frac{$ 2.84 - $ 2.31}{5 years} = \frac{$ 0.53}{5 years} = $ 0.106$ per year.

年あたり $\frac{$ 2.84 - $ 2.31}{5 年} = \frac{$ 0.53}{5 年} = $ 0.106$

Solution 2解答 2

$\frac{1}{2}$

Solution 3解答 3

$a + 7$

Solution 4解答 4

The local maximum appears to occur at $(- 1, 28),$ and the local minimum occurs at $(5, - 80) .$ The function is increasing on $(- \infty, - 1) \cup (5, \infty)$ and decreasing on $(- 1, 5) .$

極大は $(- 1, 28)$ で、極小は $(5, - 80)$ で現れるように見える。この関数は $(- \infty, - 1) \cup (5, \infty)$ で増加し、 $(- 1, 5)$ で減少する。

Graph of a polynomial with a local maximum at (-1, 28) and local minimum at (5, -80).

Solution 1解答 1

$\begin{array}{l} (f g) (x) = f (x) g (x) = (x - 1) (x^{2} - 1) = x^{3} - x^{2} - x + 1 \\ (f - g) (x) = f (x) - g (x) = (x - 1) - (x^{2} - 1) = x - x^{2} \end{array}$

No, the functions are not the same.

いいえ、この二つは同じ関数ではない。

Solution 2解答 2

A gravitational force is still a force, so $a (G (r))$ makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but $G (a (F))$ does not make sense.

重力もまた力なので、 $a (G (r))$ は太陽から距離 r にある惑星の(重力による)加速度として意味を持つ。しかし $G (a (F))$ は意味を持たない。

Solution 3解答 3

$f (g (1)) = f (3) = 3$ and $g (f (4)) = g (1) = 3$

$f (g (1)) = f (3) = 3$ と $g (f (4)) = g (1) = 3$

Solution 4解答 4

$g (f (2)) = g (5) = 3$

Solution 5解答 5

ⓐ8ⓑ20

Solution 6解答 6

$[- 4, 0) \cup (0, \infty)$

Solution 7解答 7

Possible answer:

ありうる答え:

$\begin{array}{l} g (x) = \sqrt{4 + x^{2}} \\ h (x) = \frac{4}{3 - x} \\ f = h \circ g \end{array}$

Solution 1解答 1

b (t) = h (t) + 10 = - 4.9 t^{2} + 30 t + 10

Solution 2解答 2

The graphs of $f (x)$ and $g (x)$ are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.

$f (x)$ と $g (x)$ のグラフを下に示す。この変形は横の平行移動である。関数は左へ2だけ動いている。

Graph of a square root function and a horizontally shift square foot function.

Solution 3解答 3

Solution 4解答 4

$g (x) = \frac{1}{x - 1} + 1$

Solution 5解答 5

ⓐ ⓑ

Solution 6解答 6

ⓐ $g (x) = - f (x)$ $x$ -2 0 2 4 $g (x)$ $- 5$ $- 10$ $- 15$ $- 20$ ⓑ $h (x) = f (− x)$ $x$ -2 0 2 4 $h (x)$ 15 10 5 unknown

ⓑ $h (x) = f (− x)$ $x$ -2 0 2 4 $h (x)$ 15 10 5 不明

Solution 7解答 7

Notice: $g (x) = f (- x)$ looks the same as $f (x)$ .

注意: $g (x) = f (- x)$ は $f (x)$ と同じに見える。

Solution 8解答 8

even

偶関数

Solution 9解答 9

$x$	2	4	6	8
$g (x)$	9	12	15	0

Solution 10解答 10

$g (x) = 3 x - 2$

Solution 11解答 11

$g (x) = f (\frac{1}{3} x)$ so using the square root function we get $g (x) = \sqrt{\frac{1}{3} x}$

$g (x) = f (\frac{1}{3} x)$ なので、平方根の関数を使って $g (x) = \sqrt{\frac{1}{3} x}$ となる。

Solution 1解答 1

using the variable $p$ for passing, $| p - 80 | \leq 20$

合格を表す変数 $p$ を使って $| p - 80 | \leq 20$

Solution 2解答 2

$f (x) = - | x + 2 | + 3$

Solution 3解答 3

$x = - 1$ or $x = 2$

$x = - 1$ または $x = 2$

Solution 1解答 1

$h (2) = 6$

Solution 2解答 2

Yes

はい

Solution 3解答 3

Yes

はい

Solution 4解答 4

The domain of function $f^{- 1}$ is $(- \infty, - 2)$ and the range of function $f^{- 1}$ is $(1, \infty) .$

関数 $f^{- 1}$ の定義域は $(- \infty, - 2)$ 、関数 $f^{- 1}$ の値域は $(1, \infty)$ である。

Solution 5解答 5

ⓐ $f (60) = 50.$ In 60 minutes, 50 miles are traveled.ⓑ $f^{- 1} (60) = 70.$ To travel 60 miles, it will take 70 minutes.

ⓐ $f (60) = 50.$ 60分で50マイル進む。ⓑ $f^{- 1} (60) = 70.$ 60マイル進むには70分かかる。

Solution 6解答 6

ⓐ3ⓑ5.6

Solution 7解答 7

$x = 3 y + 5$

Solution 8解答 8

$f^{- 1} (x) = {(2 - x)}^{2};$ $domain of f : [0, \infty);$ $domain of f^{- 1} : (- \infty, 2]$

式中の英語：domain：定義域

Solution 9解答 9

3.1 Section Exercises3.1 節末問題

Solution 1解答 1

A relation is a set of ordered pairs. A function is a special kind of relation in which no two ordered pairs have the same first coordinate.

関係とは順序対の集合である。関数はその特別な種類で、一つ目の座標が同じ順序対が二つ存在しないものである。

Solution 3解答 3

When a vertical line intersects the graph of a relation more than once, that indicates that for that input there is more than one output. At any particular input value, there can be only one output if the relation is to be a function.

縦線が関係のグラフと二回以上交わるということは、その入力に対して出力が二つ以上あるということである。関係が関数であるためには、どの入力値でも出力は一つしかありえない。

Solution 5解答 5

When a horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to only one input.

横線が関数のグラフと二回以上交わるということは、その出力に対して入力が二つ以上あるということである。各出力がただ一つの入力に対応するとき、その関数は一対一である。

Solution 7解答 7

function

関数

Solution 9解答 9

function

関数

Solution 11解答 11

function

関数

Solution 13解答 13

function

関数

Solution 15解答 15

function

関数

Solution 17解答 17

function

関数

Solution 19解答 19

function

関数

Solution 21解答 21

function

関数

Solution 23解答 23

function

関数

Solution 25解答 25

not a function

関数ではない

Solution 27解答 27

$f (- 3) = - 11;$
$f (2) = - 1;$
$f (- a) = - 2 a - 5;$
$- f (a) = - 2 a + 5;$
$f (a + h) = 2 a + 2 h - 5$

Solution 29解答 29

$f (- 3) = \sqrt{5} + 5;$
$f (2) = 5;$
$f (- a) = \sqrt{2 + a} + 5;$
$- f (a) = - \sqrt{2 - a} - 5;$
$f (a + h) = \sqrt{2 - a - h} + 5$

Solution 31解答 31

$f (- 3) = 2;$ $f (2) = 1 - 3 = - 2;$
$f (- a) = | - a - 1 | - | - a + 1 |;$
$- f (a) = - | a - 1 | + | a + 1 |;$
$f (a + h) = | a + h - 1 | - | a + h + 1 |$

Solution 33解答 33

$\frac{g (x) - g (a)}{x - a} = x + a + 2, x \neq a$

Solution 35解答 35

a. $f (- 2) = 14;$ b. $x = 3$

Solution 37解答 37

a. $f (5) = 10;$ b. $x = - 1$ or $x = 4$

a. $f (5) = 10;$ b. $x = - 1$ または $x = 4$

Solution 39解答 39

ⓐ $f (t) = 6 - \frac{2}{3} t;$ ⓑ $f (- 3) = 8;$ ⓒ $t = 6$

Solution 41解答 41

not a function

関数ではない

Solution 43解答 43

function

関数

Solution 45解答 45

function

関数

Solution 47解答 47

function

関数

Solution 49解答 49

function

関数

Solution 51解答 51

function

関数

Solution 53解答 53

ⓐ $f (0) = 1;$ ⓑ $f (x) = - 3, x = - 2$ or $x = 2$

ⓑ $f (x) = - 3, x = - 2$ または $x = 2$

Solution 55解答 55

not a function so it is also not a one-to-one function

関数ではないので、一対一の関数でもない

Solution 57解答 57

one-to- one function

一対一の関数

Solution 59解答 59

function, but not one-to-one

関数だが一対一ではない

Solution 61解答 61

function

関数

Solution 63解答 63

function

関数

Solution 65解答 65

not a function

関数ではない

Solution 67解答 67

$f (x) = 1, x = 2$

Solution 69解答 69

$\begin{matrix} f (- 2) = 14; & f (- 1) = 11; & f (0) = 8; & f (1) = 5; & f (2) = 2 \end{matrix}$

Solution 71解答 71

$\begin{matrix} f (- 2) = 4; & f (- 1) = 4.414; & f (0) = 4.732; & f (1) = 5; & f (2) = 5.236 \end{matrix}$

Solution 73解答 73

$\begin{matrix} f (- 2) = \frac{1}{9}; & f (- 1) = \frac{1}{3}; & f (0) = 1; & f (1) = 3; & f (2) = 9 \end{matrix}$

Solution 75解答 75

Solution 77解答 77

$[0, 100]$

Solution 79解答 79

$[- 0.001, 0 .001]$

Solution 81解答 81

$[- 1, 000, 000, 1,000,000]$

Solution 83解答 83

$[0, 10]$

Solution 85解答 85

$[−0.1, 0.1]$

Solution 87解答 87

$[- 100, 100]$

Solution 89解答 89

ⓐ $g (5000) = 50;$ ⓑThe number of cubic yards of dirt required for a garden of 100 square feet is 1.

ⓑ100平方フィートの庭に必要な土は1立方ヤードである。

Solution 91解答 91

ⓐThe height of a rocket above ground after 1 second is 200 ft.ⓑThe height of a rocket above ground after 2 seconds is 350 ft.

ⓐ打ち上げから1秒後のロケットの地上からの高さは200フィートである。ⓑ打ち上げから2秒後のロケットの地上からの高さは350フィートである。

3.2 Section Exercises3.2 節末問題

Solution 1解答 1

The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.

関数の定義域は、独立変数のどの値でその関数が定義されなくなるか、あるいは虚数になるかによって決まる。

Solution 3解答 3

There is no restriction on $x$ for $f (x) = \sqrt[3]{x}$ because you can take the cube root of any real number. So the domain is all real numbers, $(- \infty, \infty) .$ When dealing with the set of real numbers, you cannot take the square root of negative numbers. So $x$ -values are restricted for $f (x) = \sqrt[]{x}$ to nonnegative numbers and the domain is $[0, \infty) .$

どんな実数の三乗根もとれるので、 $f (x) = \sqrt[3]{x}$ では $x$ に制限が無い。よって定義域は実数全体 $(- \infty, \infty)$ である。実数の集合を扱うときは負の数の平方根をとれない。よって $f (x) = \sqrt[]{x}$ では $x$ の値が非負に限られ、定義域は $[0, \infty)$ である。

Solution 5解答 5

Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the $x$ -axis and $y$ -axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate $- \infty$ or $\infty .$ Combine the graphs to find the graph of the piecewise function.

区分けして定めた関数の各式を、それぞれ対応する定義域の上で描く。どのグラフでも $x$ 軸と $y$ 軸に同じ目盛りを使う。端点を含むところは黒丸、含まないところは白丸で示す。 $- \infty$ や $\infty$ は矢で示す。これらのグラフを合わせれば、区分けして定めた関数のグラフになる。

Solution 7解答 7

$(- \infty, \infty)$

Solution 9解答 9

$(- \infty, 3]$

Solution 11解答 11

$(- \infty, \infty)$

Solution 13解答 13

$(- \infty, \infty)$

Solution 15解答 15

$(- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, \infty)$

Solution 17解答 17

$(- \infty, - 11) \cup (- 11, 2) \cup (2, \infty)$

Solution 19解答 19

$(- \infty, - 3) \cup (- 3, 5) \cup (5, \infty)$

Solution 21解答 21

$(- \infty, 5)$

Solution 23解答 23

$[6, \infty)$

Solution 25解答 25

$(- \infty, - 9) \cup (- 9, 9) \cup (9, \infty)$

Solution 27解答 27

domain: $(2, 8],$ range $[6, 8)$

定義域: $(2, 8]$ 値域 $[6, 8)$

Solution 29解答 29

domain: $[- 4, 4],$ range: $[0, 2]$

定義域: $[- 4, 4],$ 値域: $[0, 2]$

Solution 31解答 31

domain: $[- 5, 3),$ range: $[0, 2]$

定義域: $[- 5, 3)$ 値域: $[0, 2]$

Solution 33解答 33

domain: $(- \infty, 1],$ range: $[0, \infty)$

定義域: $(- \infty, 1]$ 値域: $[0, \infty)$

Solution 35解答 35

domain: $[- 6, - \frac{1}{6}] \cup [\frac{1}{6}, 6];$ range: $[- 6, - \frac{1}{6}] \cup [\frac{1}{6}, 6]$

定義域: $[- 6, - \frac{1}{6}] \cup [\frac{1}{6}, 6];$ 値域: $[- 6, - \frac{1}{6}] \cup [\frac{1}{6}, 6]$

Solution 37解答 37

domain: $[- 3, \infty);$ range: $[0, \infty)$

定義域: $[- 3, \infty);$ 値域: $[0, \infty)$

Solution 39解答 39

domain: $(- \infty, \infty)$

定義域: $(- \infty, \infty)$

A graph displays a piecewise function with a jump discontinuity at x=1. The function is defined by a line segment ending with an open circle at (1,1) and another line segment starting with a closed circle at (1,2).

Solution 41解答 41

domain: $(- \infty, \infty)$

定義域: $(- \infty, \infty)$

Solution 43解答 43

domain: $(- \infty, \infty)$

定義域: $(- \infty, \infty)$

A piecewise graph: a parabola curving left from an open circle at (0,0), and a line segment starting from a closed circle at (0,2) extending right. It illustrates a jump discontinuity.

Solution 45解答 45

domain: $(- \infty, \infty)$

定義域: $(- \infty, \infty)$

Solution 47解答 47

$\begin{matrix} f (- 3) = 1; & f (- 2) = 0; & f (- 1) = 0; & f (0) = 0 \end{matrix}$

Solution 49解答 49

$\begin{matrix} f (- 1) = - 4; & f (0) = 6; & f (2) = 20; & f (4) = 34 \end{matrix}$

Solution 51解答 51

$\begin{matrix} f (- 1) = - 5; & f (0) = 3; & f (2) = 3; & f (4) = 16 \end{matrix}$

Solution 53解答 53

domain: $(- \infty, 1) \cup (1, \infty)$

定義域: $(- \infty, 1) \cup (1, \infty)$

Solution 55解答 55

Graph of the equation from [-0.5, -0.1].

window: $[- 0.5, - 0.1];$ range: $[4, 100]$

表示窓: $[- 0.5, - 0.1];$ 値域: $[4, 100]$

window: $[0.1, 0.5];$ range: $[4, 100]$

表示窓: $[0.1, 0.5];$ 値域: $[4, 100]$

Solution 57解答 57

$[0, 8]$

Solution 59解答 59

Many answers. One function is $f (x) = \frac{1}{\sqrt{x - 2}} .$

答えはいくつもある。一つの関数は $f (x) = \frac{1}{\sqrt{x - 2}}$ である。

Solution 61解答 61

ⓐThe fixed cost is $500.ⓑThe cost of making 25 items is $750.ⓒThe domain is [0, 100] and the range is [500, 1500].

ⓐ固定費は500ドルである。ⓑ25個作る費用は750ドルである。ⓒ定義域は [0, 100]、値域は [500, 1500] である。

3.3 Section Exercises3.3 節末問題

Solution 1解答 1

Yes, the average rate of change of all linear functions is constant.

なる。一次関数の平均変化率はどれも一定である。

Solution 3解答 3

The absolute maximum and minimum relate to the entire graph, whereas the local extrema relate only to a specific region around an open interval.

最大値と最小値はグラフ全体に関わるが、極値は開いた区間の周りの限られた範囲にしか関わらない。

Solution 5解答 5

$4 (b + 1)$

Solution 7解答 7

Solution 9解答 9

$4 x + 2 h$

Solution 11解答 11

$\frac{- 1}{13 (13 + h)}$

Solution 13解答 13

$3 h^{2} + 9 h + 9$

Solution 15解答 15

$4 x + 2 h - 3$

Solution 17解答 17

$\frac{4}{3}$

Solution 19解答 19

increasing on $(- \infty, - 2.5) \cup (1, \infty),$ decreasing on $(- 2.5, 1)$

$(- \infty, - 2.5) \cup (1, \infty)$ で増加、 $(- 2.5, 1)$ で減少

Solution 21解答 21

increasing on $(- \infty, 1) \cup (3, 4),$ decreasing on $(1, 3) \cup (4, \infty)$

$(- \infty, 1) \cup (3, 4)$ で増加、 $(1, 3) \cup (4, \infty)$ で減少

Solution 23解答 23

local maximum: $(- 3, 60),$ local minimum: $(3, - 60)$

極大: $(- 3, 60)$ 極小: $(3, - 60)$

Solution 25解答 25

absolute maximum at approximately $(7, 150),$ absolute minimum at approximately $(−7.5, −220)$

最大値はおよそ $(7, 150)$ 、最小値はおよそ $(−7.5, −220)$

Solution 27解答 27

ⓐ–3000ⓑ–1250

Solution 29解答 29

-4

Solution 31解答 31

Solution 33解答 33

–0.167

Solution 35解答 35

Local minimum at $(3, - 22),$ decreasing on $(- \infty, 3),$ increasing on $(3, \infty)$

$(3, - 22)$ で極小。 $(- \infty, 3)$ で減少、 $(3, \infty)$ で増加

Solution 37解答 37

Local minimum at $(- 2, - 2),$ decreasing on $(- 3, - 2),$ increasing on $(- 2, \infty)$

$(- 2, - 2)$ で極小。 $(- 3, - 2)$ で減少、 $(- 2, \infty)$ で増加

Solution 39解答 39

Local maximum at $(- 0.39, 5.98),$ local minima at $(- 3.15, - 47.62)$ and $(2.04, - 32.04),$ decreasing on $(- \infty, - 3.15) \cup (- 0.39, 2.04),$ increasing on $(- 3.15, - 0.39) \cup (2.04, \infty)$

$(- 0.39, 5.98)$ で極大、 $(- 3.15, - 47.62)$ と $(2.04, - 32.04)$ で極小。 $(- \infty, - 3.15) \cup (- 0.39, 2.04)$ で減少、 $(- 3.15, - 0.39) \cup (2.04, \infty)$ で増加

Solution 41解答 41

Solution 43解答 43

$b = 5$

Solution 45解答 45

2.7 gallons per minute

毎分2.7ガロン

Solution 47解答 47

approximately –0.6 milligrams per day

1日あたりおよそ –0.6 ミリグラム

3.4 Section Exercises3.4 節末問題

Solution 1解答 1

Find the numbers that make the function in the denominator $g$ equal to zero, and check for any other domain restrictions on $f$ and $g,$ such as an even-indexed root or zeros in the denominator.

分母の関数 $g$ を0にする数を求め、さらに $f$ と $g$ に偶数乗根や分母の0といった別の定義域の制限が無いかを確かめる。

Solution 3解答 3

Yes. Sample answer: Let $f (x) = x + 1 and g (x) = x - 1.$ Then $f (g (x)) = f (x - 1) = (x - 1) + 1 = x$ and $g (f (x)) = g (x + 1) = (x + 1) - 1 = x .$ So $f \circ g = g \circ f .$

ある。答えの例: $f (x) = x + 1 および g (x) = x - 1.$ とする。すると $f (g (x)) = f (x - 1) = (x - 1) + 1 = x$ かつ $g (f (x)) = g (x + 1) = (x + 1) - 1 = x$ である。よって $f \circ g = g \circ f$

Solution 5解答 5

$(f + g) (x) = 2 x + 6,$ domain: $(- \infty, \infty)$

$(f + g) (x) = 2 x + 6$ 定義域: $(- \infty, \infty)$

$(f - g) (x) = 2 x^{2} + 2 x - 6,$ domain: $(- \infty, \infty)$

$(f - g) (x) = 2 x^{2} + 2 x - 6$ 定義域: $(- \infty, \infty)$

$(f g) (x) = - x^{4} - 2 x^{3} + 6 x^{2} + 12 x,$ domain: $(- \infty, \infty)$

$(f g) (x) = - x^{4} - 2 x^{3} + 6 x^{2} + 12 x$ 定義域: $(- \infty, \infty)$

$(\frac{f}{g}) (x) = \frac{x^{2} + 2 x}{6 - x^{2}},$ domain: $(- \infty, - \sqrt{6}) \cup (- \sqrt{6}, \sqrt{6}) \cup (\sqrt{6}, \infty)$

$(\frac{f}{g}) (x) = \frac{x^{2} + 2 x}{6 - x^{2}}$ 定義域: $(- \infty, - \sqrt{6}) \cup (- \sqrt{6}, \sqrt{6}) \cup (\sqrt{6}, \infty)$

Solution 7解答 7

$(f + g) (x) = \frac{4 x^{3} + 8 x^{2} + 1}{2 x},$ domain: $(- \infty, 0) \cup (0, \infty)$

$(f + g) (x) = \frac{4 x^{3} + 8 x^{2} + 1}{2 x}$ 定義域: $(- \infty, 0) \cup (0, \infty)$

$(f - g) (x) = \frac{4 x^{3} + 8 x^{2} - 1}{2 x},$ domain: $(- \infty, 0) \cup (0, \infty)$

$(f - g) (x) = \frac{4 x^{3} + 8 x^{2} - 1}{2 x}$ 定義域: $(- \infty, 0) \cup (0, \infty)$

$(f g) (x) = x + 2,$ domain: $(- \infty, 0) \cup (0, \infty)$

$(f g) (x) = x + 2$ 定義域: $(- \infty, 0) \cup (0, \infty)$

$(\frac{f}{g}) (x) = 4 x^{3} + 8 x^{2},$ domain: $(- \infty, 0) \cup (0, \infty)$

$(\frac{f}{g}) (x) = 4 x^{3} + 8 x^{2}$ 定義域: $(- \infty, 0) \cup (0, \infty)$

Solution 9解答 9

$(f + g) (x) = 3 x^{2} + \sqrt{x - 5},$ domain: $[5, \infty)$

$(f + g) (x) = 3 x^{2} + \sqrt{x - 5}$ 定義域: $[5, \infty)$

$(f - g) (x) = 3 x^{2} - \sqrt{x - 5},$ domain: $[5, \infty)$

$(f - g) (x) = 3 x^{2} - \sqrt{x - 5}$ 定義域: $[5, \infty)$

$(f g) (x) = 3 x^{2} \sqrt{x - 5},$ domain: $[5, \infty)$

$(f g) (x) = 3 x^{2} \sqrt{x - 5}$ 定義域: $[5, \infty)$

$(\frac{f}{g}) (x) = \frac{3 x^{2}}{\sqrt{x - 5}},$ domain: $(5, \infty)$

$(\frac{f}{g}) (x) = \frac{3 x^{2}}{\sqrt{x - 5}}$ 定義域: $(5, \infty)$

Solution 11解答 11

ⓐ $3 = 2 (9 x^{2} - 30 x + 25) + 1 = 18 x^{2} - 60 x + 51$ ⓑ $f (g (x)) = 2 {(3 x - 5)}^{2} + 1$ ⓒ $g (f (x)) = 3 (2 x^{2} + 1) - 5 = 6 x^{2} - 2$ ⓓ $(g \circ g) (x) = 3 (3 x - 5) - 5 = 9 x - 20$ ⓔ $(f \circ f) (- 2) = 163$

Solution 13解答 13

$f (g (x)) = \sqrt{x^{2} + 3} + 2, g (f (x)) = x + 4 \sqrt{x} + 7$

Solution 15解答 15

$f (g (x)) = \sqrt[3]{\frac{x + 1}{x^{3}}} = \frac{\sqrt[3]{x + 1}}{x}, g (f (x)) = \frac{\sqrt[3]{x} + 1}{x}$

Solution 17解答 17

$(f \circ g) (x) = \frac{1}{\frac{2}{x} + 4 - 4} = \frac{x}{2}, (g \circ f) (x) = 2 x - 4$

Solution 19解答 19

$f (g (h (x))) = {(\frac{1}{x + 3})}^{2} + 1$

Solution 21解答 21

ⓐ $(g \circ f) (x) = - \frac{3}{\sqrt{2 - 4 x}}$ ⓑ $(- \infty, \frac{1}{2})$

Solution 23解答 23

ⓐ $(0, 2) \cup (2, \infty);$ ⓑ $(- \infty, - 2) \cup (2, \infty);$ ⓒ $(0, \infty)$

Solution 25解答 25

$(1, \infty)$

Solution 27解答 27

sample: $\begin{array}{l} f (x) = x^{3} \\ g (x) = x - 5 \end{array}$

例: $\begin{array}{l} f (x) = x^{3} \\ g (x) = x - 5 \end{array}$

Solution 29解答 29

sample: $\begin{array}{l} f (x) = \frac{4}{x} \\ g (x) = {(x + 2)}^{2} \end{array}$

例: $\begin{array}{l} f (x) = \frac{4}{x} \\ g (x) = {(x + 2)}^{2} \end{array}$

Solution 31解答 31

sample: $\begin{array}{l} f (x) = \sqrt[3]{x} \\ g (x) = \frac{1}{2 x - 3} \end{array}$

例: $\begin{array}{l} f (x) = \sqrt[3]{x} \\ g (x) = \frac{1}{2 x - 3} \end{array}$

Solution 33解答 33

sample: $\begin{array}{l} f (x) = \sqrt[4]{x} \\ g (x) = \frac{3 x - 2}{x + 5} \end{array}$

例: $\begin{array}{l} f (x) = \sqrt[4]{x} \\ g (x) = \frac{3 x - 2}{x + 5} \end{array}$

Solution 35解答 35

sample: $\begin{array}{l} f (x) = \sqrt{x} \\ g (x) = 2 x + 6 \end{array}$

例: $\begin{array}{l} f (x) = \sqrt{x} \\ g (x) = 2 x + 6 \end{array}$

Solution 37解答 37

sample: $\begin{array}{l} f (x) = \sqrt[3]{x} \\ g (x) = (x - 1) \end{array}$

例: $\begin{array}{l} f (x) = \sqrt[3]{x} \\ g (x) = (x - 1) \end{array}$

Solution 39解答 39

sample: $\begin{array}{l} f (x) = x^{3} \\ g (x) = \frac{1}{x - 2} \end{array}$

例: $\begin{array}{l} f (x) = x^{3} \\ g (x) = \frac{1}{x - 2} \end{array}$

Solution 41解答 41

sample: $\begin{array}{l} f (x) = \sqrt{x} \\ g (x) = \frac{2 x - 1}{3 x + 4} \end{array}$

例: $\begin{array}{l} f (x) = \sqrt{x} \\ g (x) = \frac{2 x - 1}{3 x + 4} \end{array}$

Solution 43解答 43

Solution 45解答 45

Solution 47解答 47

Solution 49解答 49

Solution 51解答 51

Solution 53解答 53

Solution 55解答 55

Solution 57解答 57

Solution 59解答 59

Solution 61解答 61

Solution 63解答 63

Solution 65解答 65

Solution 67解答 67

Solution 69解答 69

Solution 71解答 71

Solution 73解答 73

$f (g (0)) = 27, g (f (0)) = - 94$

Solution 75解答 75

$f (g (0)) = \frac{1}{5}, g (f (0)) = 5$

Solution 77解答 77

$18 x^{2} + 60 x + 51$

Solution 79解答 79

$g \circ g (x) = 9 x + 20$

Solution 81解答 81

Solution 83解答 83

$(- \infty, \infty)$

Solution 85解答 85

False

誤り

Solution 87解答 87

$(f \circ g) (6) = 6$ ; $(g \circ f) (6) = 6$

Solution 89解答 89

$(f \circ g) (11) = 11, (g \circ f) (11) = 11$

Solution 91解答 91

Solution 93解答 93

$A (t) = π {(25 \sqrt{t + 2})}^{2}$ and $A (2) = π {(25 \sqrt{4})}^{2} = 2500 π$ square inches

$A (t) = π {(25 \sqrt{t + 2})}^{2}$ および $A (2) = π {(25 \sqrt{4})}^{2} = 2500 π$ 平方インチ

Solution 95解答 95

$A (5) = π {(2 (5) + 1)}^{2} = 121 π$ square units

$A (5) = π {(2 (5) + 1)}^{2} = 121 π$ 平方単位

Solution 97解答 97

ⓐ $N (T (t)) = 23 {(5 t + 1.5)}^{2} - 56 (5 t + 1.5) + 1$ ⓑ3.38 hours

ⓑ3.38時間

3.5 Section Exercises3.5 節末問題

Solution 1解答 1

A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.

入力に定数を足したり引いたりすると横の平行移動になる。出力に定数を足したり引いたりすると縦の平行移動になる。

Solution 3解答 3

A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output.

入力に1より大きい定数を掛けると横の縮小になる。出力に0と1のあいだの定数を掛けると縦の縮小になる。

Solution 5解答 5

For a function $f,$ substitute $(- x)$ for $(x)$ in $f (x) .$ Simplify. If the resulting function is the same as the original function, $f (- x) = f (x),$ then the function is even. If the resulting function is the opposite of the original function, $f (- x) = - f (x),$ then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

関数 $f$ について、 $f (x)$ の $(x)$ に $(- x)$ を代入して簡単にする。できた関数がもとの関数と同じ、すなわち $f (- x) = f (x)$ なら偶関数である。できた関数がもとの関数の反数、すなわち $f (- x) = - f (x)$ ならもとの関数は奇関数である。同じでも反数でもなければ、奇関数でも偶関数でもない。

Solution 7解答 7

$g (x) = | x - 1 | - 3$

Solution 9解答 9

$g (x) = \frac{1}{{(x + 4)}^{2}} + 2$

Solution 11解答 11

The graph of $f (x + 43)$ is a horizontal shift to the left 43 units of the graph of $f .$

$f (x + 43)$ のグラフは、 $f$ のグラフを左へ43だけ横に動かしたものである。

Solution 13解答 13

The graph of $f (x - 4)$ is a horizontal shift to the right 4 units of the graph of $f .$

$f (x - 4)$ のグラフは、 $f$ のグラフを右へ4だけ横に動かしたものである。

Solution 15解答 15

The graph of $f (x) + 8$ is a vertical shift up 8 units of the graph of $f .$

$f (x) + 8$ のグラフは、 $f$ のグラフを上へ8だけ縦に動かしたものである。

Solution 17解答 17

The graph of $f (x) - 7$ is a vertical shift down 7 units of the graph of $f .$

$f (x) - 7$ のグラフは、 $f$ のグラフを下へ7だけ縦に動かしたものである。

Solution 19解答 19

The graph of $f (x + 4) - 1$ is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of $f .$

$f (x + 4) - 1$ のグラフは、 $f$ のグラフを左へ4だけ横に、下へ1だけ縦に動かしたものである。

Solution 21解答 21

decreasing on $(- \infty, - 3)$ and increasing on $(- 3, \infty)$

$(- \infty, - 3)$ で減少、 $(- 3, \infty)$ で増加

Solution 23解答 23

decreasing on $(0, \infty)$

$(0, \infty)$ で減少

Solution 25解答 25

A graph displays an exponential function h, characterized by a left-to-right increasing curve. It approaches a horizontal asymptote at y=-3 for negative x, and rises steeply for positive x.

Solution 27解答 27

Solution 29解答 29

Solution 31解答 31

$g (x) = f (x - 1), h (x) = f (x) + 1$

Solution 33解答 33

$f (x) = | x - 3 | - 2$

Solution 35解答 35

$f (x) = \sqrt{x + 3} - 1$

Solution 37解答 37

$f (x) = {(x - 2)}^{2}$

Solution 39解答 39

$f (x) = | x + 3 | - 2$

Solution 41解答 41

$f (x) = - \sqrt{x}$

Solution 43解答 43

$f (x) = - {(x + 1)}^{2} + 2$

Solution 45解答 45

$f (x) = \sqrt{- x} + 1$

Solution 47解答 47

even

偶関数

Solution 49解答 49

odd

奇関数

Solution 51解答 51

even

偶関数

Solution 53解答 53

The graph of $g$ is a vertical reflection (across the $x$ -axis) of the graph of $f .$

$g$ のグラフは、 $f$ のグラフを縦に( $x$ 軸に関して)折り返したものである。

Solution 55解答 55

The graph of $g$ is a vertical stretch by a factor of 4 of the graph of $f .$

$g$ のグラフは、 $f$ のグラフを縦に4倍に伸ばしたものである。

Solution 57解答 57

The graph of $g$ is a horizontal compression by a factor of $\frac{1}{5}$ of the graph of $f .$

$g$ のグラフは、 $f$ のグラフを横に $\frac{1}{5}$ 倍に縮めたものである。

Solution 59解答 59

The graph of $g$ is a horizontal stretch by a factor of 3 of the graph of $f .$

$g$ のグラフは、 $f$ のグラフを横に3倍に伸ばしたものである。

Solution 61解答 61

The graph of $g$ is a horizontal reflection across the $y$ -axis and a vertical stretch by a factor of 3 of the graph of $f .$

$g$ のグラフは、 $f$ のグラフを $y$ 軸に関して横に折り返し、縦に3倍に伸ばしたものである。

Solution 63解答 63

$g (x) = | - 4 x |$

Solution 65解答 65

$g (x) = \frac{1}{3 {(x + 2)}^{2}} - 3$

Solution 67解答 67

$g (x) = \frac{1}{2} {(x - 5)}^{2} + 1$

Solution 69解答 69

The graph of the function $f (x) = x^{2}$ is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.

関数 $f (x) = x^{2}$ のグラフを左へ1だけ動かし、縦に4倍に伸ばし、下へ5だけ動かしたものである。

Solution 71解答 71

The graph of $f (x) = | x |$ is stretched vertically by a factor of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, and then shifted vertically 3 units up.

$f (x) = | x |$ のグラフを縦に2倍に伸ばし、横に右へ4だけ動かし、横の軸に関して折り返し、そのうえで縦に上へ3だけ動かしたものである。

Solution 73解答 73

The graph of the function $f (x) = x^{3}$ is compressed vertically by a factor of $\frac{1}{2} .$

関数 $f (x) = x^{3}$ のグラフを縦に $\frac{1}{2}$ 倍に縮めたものである。

Solution 75解答 75

The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.

この関数のグラフを横に3倍に伸ばし、そのうえで縦に下へ3だけ動かしたものである。

Solution 77解答 77

The graph of $f (x) = \sqrt{x}$ is reflected across the y-axis and then shifted right 4 units.

$f (x) = \sqrt{x}$ のグラフを y軸に関して折り返し、そのうえで右へ4だけ動かしたものである。

Solution 79解答 79

Solution 81解答 81

3.6 Section Exercises3.6 節末問題

Solution 1解答 1

Isolate the absolute value term so that the equation is of the form $| A | = B .$ Form one equation by setting the expression inside the absolute value symbol, $A,$ equal to the expression on the other side of the equation, $B .$ Form a second equation by setting $A$ equal to the opposite of the expression on the other side of the equation, $- B .$ Solve each equation for the variable.

方程式が $| A | = B$ の形になるように絶対値の項だけを残す。絶対値の記号の中の式 $A$ を、方程式のもう一方の側の式 $B$ と等しいとおいて一つ目の方程式を作る。次に $A$ を、もう一方の側の式の反数 $- B$ と等しいとおいて二つ目の方程式を作る。それぞれの方程式を変数について解く。

Solution 3解答 3

The graph of the absolute value function does not cross the $x$ -axis, so the graph is either completely above or completely below the $x$ -axis.

絶対値関数のグラフが $x$ 軸と交わらないので、グラフは完全に $x$ 軸の上か、完全に下かのどちらかである。

Solution 5解答 5

The distance from x to 8 can be represented using the absolute value statement: ∣ x − 8 ∣ = 4.

x から8までの距離は、絶対値の文 ∣ x − 8 ∣ = 4 で表せる。

Solution 7解答 7

∣ x − 10 ∣ ≥ 15

Solution 9解答 9

There are no x-intercepts.

x切片は無い。

Solution 11解答 11

(−4, 0) and (2, 0)

(−4, 0) と (2, 0)

Solution 13解答 13

$(0, - 4), (4, 0), (- 2, 0)$

Solution 15解答 15

$(0, 7), (25, 0), (- 7, 0)$

Solution 17解答 17

A V-shaped graph with a vertex at (-1, 0), showing points (-3, 2), (-2, 1), (0, 1), and (1, 2).

Solution 19解答 19

A V-shaped graph is displayed on a Cartesian coordinate system. The x-axis ranges from -5 to 4, and the y-axis ranges from -3 to 5. The vertex of the graph is at the point (0, -2). The graph opens upwards, passing through the x-axis at (-2, 0) and (2, 0). The line segment for x < 0 has a slope of -1, and the line segment for x > 0 has a slope of 1. Arrows on both ends indicate that the graph extends indefinitely, representing the function y = |x| - 2.

Solution 21解答 21

A graph showing a V-shaped function on a Cartesian coordinate system, with its vertex located at (0, -2).

Solution 23解答 23

A graph on a coordinate plane displays a V-shaped function. The vertex of the V is located at the point (0, -2). From this vertex, one arm of the V extends downwards and to the left, passing through approximately (-2, -4), and the other arm extends downwards and to the right, passing through approximately (2, -4). Both arms are indicated with arrows, signifying that the function continues indefinitely in those directions. The coordinate plane shows x-axis values from -5 to 5 and y-axis values from -7 to 5, with grid lines at each integer increment.

Solution 25解答 25

A graph showing an absolute value function in a V-shape. The vertex of the function is located at the point (-3, 1), and the graph opens upwards on a coordinate plane.

Solution 27解答 27

A graph displays an absolute value function, forming a V-shape on a Cartesian coordinate plane. The x-axis is labeled from -6 to 5, and the y-axis is labeled from -6 to 6. The graph has its vertex at (2, -3) and opens upwards, passing through points such as (0, 1), (1, -1), (3, -1), and (4, 1).

Solution 29解答 29

A coordinate plane shows an absolute value function graphed as a V-shape opening downwards. The x-axis ranges from -6 to 5, and the y-axis ranges from -6 to 6. The vertex of the V-shape is located at the point (1, -3). The two linear segments extend downwards and outwards from the vertex, with arrows indicating that the graph continues indefinitely.

Solution 31解答 31

A graph on a coordinate plane shows a V-shaped function. The vertex of the V is located at the point (-4, -3). From this vertex, two arms extend upwards, with arrows indicating they continue infinitely.

Solution 33解答 33

range: $[- 400, 100]$

値域: $[- 400, 100]$

Solution 35解答 35

Solution 37解答 37

There is no solution for $a$ that will keep the function from having a $y$ -intercept. The absolute value function always crosses the $y$ -intercept when $x = 0.$

この関数に $y$ 切片を持たせないような $a$ の解は無い。絶対値関数は $x = 0.$ のときつねに $y$ 切片と交わる。

Solution 39解答 39

$| p - 0.08 | \leq 0.015$

Solution 41解答 41

$| x - 5.0 | \leq 0.01$

3.7 Section Exercises3.7 節末問題

Solution 1解答 1

Each output of a function must have exactly one output for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that $y$ -values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no $y$ -values repeat and the function is one-to-one.

関数が一対一であるためには、各出力にちょうど一つの入力が対応しなければならない。横線が関数のグラフと二回以上交わるなら、 $y$ の値が繰り返されており、その関数は一対一ではない。関数のグラフと二回以上交わる横線が無いなら、 $y$ の値は繰り返されず、その関数は一対一である。

Solution 3解答 3

Yes. For example, $f (x) = \frac{1}{x}$ is its own inverse.

ある。たとえば $f (x) = \frac{1}{x}$ は自分自身の逆関数である。

Solution 5解答 5

Given a function $y = f (x),$ solve for $x$ in terms of $y .$ Interchange the $x$ and $y .$ Solve the new equation for $y .$ The expression for $y$ is the inverse, $y = f^{- 1} (x) .$

関数 $y = f (x)$ が与えられたら、 $y$ を使って $x$ について解く。 $x$ と $y$ を入れ替える。新しい方程式を $y$ について解く。その $y$ の式が逆関数 $y = f^{- 1} (x)$ である。

Solution 7解答 7

$f^{- 1} (x) = x - 3$

Solution 9解答 9

$f^{- 1} (x) = 2 - x$

Solution 11解答 11

$f^{- 1} (x) = \frac{- 2 x}{x - 1}$

Solution 13解答 13

domain of $f (x) : [- 7, \infty); f^{- 1} (x) = \sqrt{x} - 7$

$f (x) : [- 7, \infty); f^{- 1} (x) = \sqrt{x} - 7$ の定義域

Solution 15解答 15

domain of $f (x) : [0, \infty); f^{- 1} (x) = \sqrt{x + 5}$

$f (x) : [0, \infty); f^{- 1} (x) = \sqrt{x + 5}$ の定義域

Solution 16解答 16

ⓐ $f (g (x)) = x$ and $g (f (x)) = x .$ ⓑThis tells us that $f$ and $g$ are inverse functions

ⓐ $f (g (x)) = x$ と $g (f (x)) = x$ ⓑこれは $f$ と $g$ が互いに逆関数であることを教えてくれる。

Solution 17解答 17

$f (g (x)) = x, g (f (x)) = x$

Solution 19解答 19

one-to-one

一対一

Solution 21解答 21

one-to-one

一対一

Solution 23解答 23

not one-to-one

一対一ではない

Solution 25解答 25

$3$

Solution 27解答 27

$2$

Solution 29解答 29

Graph of a square root function and its inverse.

Solution 31解答 31

$[2, 10]$

Solution 33解答 33

$6$

Solution 35解答 35

$- 4$

Solution 37解答 37

$0$

Solution 39解答 39

$1$

Solution 41解答 41

$x$	1	4	7	12	16
$f^{- 1} (x)$	3	6	9	13	14

Solution 43解答 43

$f^{- 1} (x) = {(1 + x)}^{1 / 3}$

Graph of a cubic function and its inverse.

Solution 45解答 45

$f^{- 1} (x) = \frac{5}{9} (x - 32) .$ Given the Fahrenheit temperature, $x,$ this formula allows you to calculate the Celsius temperature.

$f^{- 1} (x) = \frac{5}{9} (x - 32)$ 華氏の気温 $x$ が与えられたとき、この公式で摂氏の気温を計算できる。

Solution 47解答 47

$t (d) = \frac{d}{50},$ $t (180) = \frac{180}{50} .$ The time for the car to travel 180 miles is 3.6 hours.

$t (d) = \frac{d}{50}$ $t (180) = \frac{180}{50}$ 自動車が180マイル走るのにかかる時間は3.6時間である。

Review Exercises復習問題

Solution 1解答 1

function

関数

Solution 3解答 3

not a function

関数ではない

Solution 5解答 5

$f (- 3) = - 27;$ $f (2) = - 2;$ $f (- a) = - 2 a^{2} - 3 a;$
$- f (a) = 2 a^{2} - 3 a;$ $f (a + h) = - 2 a^{2} + 3 a - 4 a h + 3 h - 2 h^{2}$

Solution 7解答 7

one-to-one

一対一

Solution 9解答 9

function

関数

Solution 11解答 11

function

関数

Solution 13解答 13

A graph of an upward-opening parabola with its vertex at (0, -2) and passing through approximately (-1.4, 0) and (1.4, 0). The curve extends upwards on both sides.

Solution 15解答 15

$2$

Solution 17解答 17

$x = - 1.8$ or $or x = 1.8$

$x = - 1.8$ または $または x = 1.8$

Solution 19解答 19

$\frac{- 64 + 80 a - 16 a^{2}}{- 1 + a} = - 16 a + 64$

Solution 21解答 21

$(- \infty, - 2) \cup (- 2, 6) \cup (6, \infty)$

Solution 23解答 23

A piecewise linear graph featuring a jump discontinuity at x = -2. The function approaches -1 from the left with an open circle, and has a value of 1 from the right, continuing downwards.

Solution 25解答 25

$31$

Solution 27解答 27

increasing $(2, \infty);$ decreasing $(- \infty, 2)$

$(2, \infty);$ で増加、 $(- \infty, 2)$ で減少

Solution 29解答 29

increasing $(- 3, 1);$ constant $(- \infty, - 3) \cup (1, \infty)$

$(- 3, 1);$ で増加、 $(- \infty, - 3) \cup (1, \infty)$ で一定

Solution 31解答 31

local minimum $(- 2, - 3);$ local maximum $(1, 3)$

極小 $(- 2, - 3);$ 極大 $(1, 3)$

Solution 33解答 33

$(- 1.8, 10)$

Solution 35解答 35

$(f \circ g) (x) = 17 - 18 x; (g \circ f) (x) = - 7 - 18 x$

Solution 37解答 37

$(f \circ g) (x) = \sqrt{\frac{1}{x} + 2}; (g \circ f) (x) = \frac{1}{\sqrt{x + 2}}$

Solution 39解答 39

$(f \circ g) (x) = \frac{1 + x}{1 + 4 x}, x \neq 0, x \neq - \frac{1}{4}$

Solution 41解答 41

$(f \circ g) (x) = \frac{1}{\sqrt{x}}, x > 0$

Solution 43解答 43

sample: $g (x) = \frac{2 x - 1}{3 x + 4}; f (x) = \sqrt{x}$

例: $g (x) = \frac{2 x - 1}{3 x + 4}; f (x) = \sqrt{x}$

Solution 45解答 45

Solution 47解答 47

Solution 49解答 49

Solution 51解答 51

A V-shaped graph resembling an absolute value function is shown on a Cartesian plane. Its vertex is at (2, -24), and it crosses the x-axis at (-4, 0) and (8, 0).

Solution 53解答 53

Solution 55解答 55

$f (x) = | x - 3 |$

Solution 57解答 57

even

偶関数

Solution 59解答 59

odd

奇関数

Solution 61解答 61

even

偶関数

Solution 63解答 63

$f (x) = \frac{1}{2} | x + 2 | + 1$

Solution 65解答 65

$f (x) = - 3 | x - 3 | + 3$

Solution 67解答 67

A graph displays a blue V-shaped function, symmetrical around x=3, with its vertex at (3, 0). The two linear segments extend downwards, passing through points (0, -3) and (6, -3).

Solution 69解答 69

$f^{- 1} (x) = \frac{x - 9}{10}$

Solution 71解答 71

$f^{- 1} (x) = \sqrt{x - 1}$

Solution 73解答 73

The function is one-to-one.

この関数は一対一である。

A graph of the reciprocal function y = 1/x, showing a curve in the first and third quadrants of the Cartesian coordinate system. The x-axis and y-axis are labeled with tick marks from -4 to 4. The curve has a vertical asymptote at x=0 (the y-axis) and a horizontal asymptote at y=0 (the x-axis). In the first quadrant, the curve starts high near the positive y-axis, passes through (1,1) and extends towards the positive x-axis. In the third quadrant, the curve starts low near the negative y-axis, passes through (-1,-1) and extends towards the negative x-axis.

Solution 75解答 75

$5$

Practice Test実力テスト

Solution 1解答 1

The relation is a function.

この関係は関数である。

Solution 3解答 3

−16

Solution 5解答 5

The graph is a parabola and the graph fails the horizontal line test.

グラフは放物線であり、横線の判定法を通らない。

Solution 7解答 7

$2 a^{2} - a$

Solution 9解答 9

$- 2 (a + b) + 1$

Solution 11解答 11

$\sqrt{2}$

Solution 13解答 13

A graph displays a curve resembling a square root function, starting at approximately (-6, -1) and extending upward and to the right, passing through (-5, 0).

Solution 15解答 15

$even$

式中の英語：even：偶数

Solution 17解答 17

$odd$

式中の英語：odd：奇数

Solution 19解答 19

$f^{- 1} (x) = \frac{x + 5}{3}$

Solution 21解答 21

$(- \infty, - 1.1) and (1.1, \infty)$

式中の英語：and：および

Solution 23解答 23

$(1.1, - 0.9)$

Solution 25解答 25

$f (2) = 2$

Solution 27解答 27

$f (x) = {\begin{matrix} | x | if x \leq 2 \\ 3 if x > 2 \end{matrix}$

Solution 29解答 29

$x = 2$

Solution 31解答 31

yes

はい

Solution 33解答 33

$f^{- 1} (x) = - \frac{x - 11}{2}$