第1章関数とグラフ

Typically, a function is represented using one or more of the following tools:

We can identify a function in each form, but we can also use them together. For instance, we can plot on a graph the values from a table or create a table from a formula.

Functions described using a table of values arise frequently in real-world applications. Consider the following simple example. We can describe temperature on a given day as a function of time of day. Suppose we record the temperature every hour for a 24-hour period starting at midnight. We let our input variable $x$ be the time after midnight, measured in hours, and the output variable $y$ be the temperature $x$ hours after midnight, measured in degrees Fahrenheit. We record our data in Table 1.1.

We can see from the table that temperature is a function of time, and the temperature decreases, then increases, and then decreases again. However, we cannot get a clear picture of the behavior of the function without graphing it.

Given a function $f$ described by a table, we can provide a visual picture of the function in the form of a graph. Graphing the temperatures listed in Table 1.1 can give us a better idea of their fluctuation throughout the day. Figure 1.6 shows the plot of the temperature function.

From the points plotted on the graph in Figure 1.6, we can visualize the general shape of the graph. It is often useful to connect the dots in the graph, which represent the data from the table. In this example, although we cannot make any definitive conclusion regarding what the temperature was at any time for which the temperature was not recorded, given the number of data points collected and the pattern in these points, it is reasonable to suspect that the temperatures at other times followed a similar pattern, as we can see in Figure 1.7.

Sometimes we are not given the values of a function in table form, rather we are given the values in an explicit formula. Formulas arise in many applications. For example, the area of a circle of radius $r$ is given by the formula $A\left(r\right)=\pi r^2.$ When an object is thrown upward from the ground with an initial velocity $v_0$ ft/s, its height above the ground from the time it is thrown until it hits the ground is given by the formula $s\left(t\right)=\mathrm{-16}{t}^2+v_{0}t.$ When $P$ dollars are invested in an account at an annual interest rate $r$ compounded continuously, the amount of money after $t$ years is given by the formula $A\left(t\right)=P{e}^{rt}.$ Algebraic formulas are important tools to calculate function values. Often we also represent these functions visually in graph form.

Given an algebraic formula for a function $f,$ the graph of $f$ is the set of points $\left(x,f\left(x\right)\right),$ where $x$ is in the domain of $f$ and $f(x)$ is in the range. To graph a function given by a formula, it is helpful to begin by using the formula to create a table of inputs and outputs. If the domain of $f$ consists of an infinite number of values, we cannot list all of them, but because listing some of the inputs and outputs can be very useful, it is often a good way to begin.

When creating a table of inputs and outputs, we typically check to determine whether zero is an output. Those values of $x$ where $f\left(x\right)=0$ are called the zeros of a function. For example, the zeros of $f\left(x\right)=x^2-4$ are $x=\pm 2.$ The zeros determine where the graph of $f$ intersects the $x$-axis, which gives us more information about the shape of the graph of the function. The graph of a function may never intersect the x-axis, or it may intersect multiple (or even infinitely many) times.

Another point of interest is the $y$-intercept, if it exists. The $y$-intercept is given by $\left(0,f\left(0\right)\right).$

Since a function has exactly one output for each input, the graph of a function can have, at most, one $y$-intercept. If $x=0$ is in the domain of a function $f,$ then $f$ has exactly one $y$-intercept. If $x=0$ is not in the domain of $f,$ then $f$ has no $y$-intercept. Similarly, for any real number $c,$ if $c$ is in the domain of $f,$ there is exactly one output $f\left(c\right),$ and the line $x=c$ intersects the graph of $f$ exactly once. On the other hand, if $c$ is not in the domain of $f,f(c)$ is not defined and the line $x=c$ does not intersect the graph of $f.$ This property is summarized in the vertical line test.

We can use this test to determine whether a set of plotted points represents the graph of a function (Figure 1.8).

Note that for this function and the function $f\left(x\right)=\mathrm{-4}x+2$ graphed in Figure 1.9, the values of $f(x)$ are getting smaller as $x$ is getting larger. A function with this property is said to be decreasing. On the other hand, for the function $f\left(x\right)=\sqrt{x+3}+1$ graphed in Figure 1.10, the values of $f(x)$ are getting larger as the values of $x$ are getting larger. A function with this property is said to be increasing. It is important to note, however, that a function can be increasing on some interval or intervals and decreasing over a different interval or intervals. For example, using our temperature function in Figure 1.6, we can see that the function is decreasing on the interval $(0,4),$ increasing on the interval $(4,14),$ and then decreasing on the interval $\left(14,23\right).$ We make the idea of a function increasing or decreasing over a particular interval more precise in the next definition.

For example, the function $f(x)=3x$ is increasing on the interval $(\text{−}\infty ,\infty )$ because $3x_1<3x_2$ whenever $x_1<x_2.$ On the other hand, the function $f\left(x\right)=\text{−}{x}^3$ is strictly decreasing on the interval $(\text{−}\infty ,\infty )$ because $\text{−}{x}_1^3>-x_2^3$ whenever $x_1<x_2$ (Figure 1.11).