1.2 Basic Classes of Functions

Linear Functions and Slope

The easiest type of function to consider is a linear function. Linear functions have the form $f(x)=ax+b,$ where $a$ and $b$ are constants. In Figure 1.15, we see examples of linear functions when $a$ is positive, negative, and zero. Note that if $a>0,$ the graph of the line rises as $x$ increases. In other words, $f(x)=ax+b$ is increasing on $\text{(−∞, ∞)}.$ If $a<0,$ the graph of the line falls as $x$ increases. In this case, $f(x)=ax+b$ is decreasing on $\text{(−∞, ∞)}.$ If $a=0,$ the line is horizontal.

Figure 1.15 These linear functions are increasing or decreasing on $\text{(-∞, ∞)}$ and one function is a horizontal line.

As suggested by Figure 1.15, the graph of any linear function is a line. One of the distinguishing features of a line is its slope. The slope is the change in $y$ for each unit change in $x.$ The slope measures both the steepness and the direction of a line. If the slope is positive, the line points upward when moving from left to right. If the slope is negative, the line points downward when moving from left to right. If the slope is zero, the line is horizontal. To calculate the slope of a line, we need to determine the ratio of the change in $y$ versus the change in $x.$ To do so, we choose any two points $(x_1,y_1)$ and $(x_2,y_2)$ on the line and calculate $\frac{y_2-y_1}{x_2-x_1}.$ In Figure 1.16, we see this ratio is independent of the points chosen.

Figure 1.16 For any linear function, the slope $(y_2-y_1)\text{/}(x_2-x_1)$ is independent of the choice of points $(x_1,y_1)$ and $(x_2,y_2)$ on the line.

We now examine the relationship between slope and the formula for a linear function. Consider the linear function given by the formula $f\left(x\right)=ax+b.$ As discussed earlier, we know the graph of a linear function is given by a line. We can use our definition of slope to calculate the slope of this line. As shown, we can determine the slope by calculating $(y_2-y_1)\text{/}(x_2-x_1)$ for any points $(x_1,y_1)$ and $(x_2,y_2)$ on the line. Evaluating the function $f$ at $x=0,$ we see that $(0,b)$ is a point on this line. Evaluating this function at $x=1,$ we see that $(1,a+b)$ is also a point on this line. Therefore, the slope of this line is

We have shown that the coefficient $a$ is the slope of the line. We can conclude that the formula $f(x)=ax+b$ describes a line with slope $a.$ Furthermore, because this line intersects the $y$-axis at the point $(0,b),$ we see that the $y$-intercept for this linear function is $(0,b).$ We conclude that the formula $f(x)=ax+b$ tells us the slope, $a,$ and the $y$-intercept, $(0,b),$ for this line. Since we often use the symbol $m$ to denote the slope of a line, we can write

to denote the slope-intercept form of a linear function.

Sometimes it is convenient to express a linear function in different ways. For example, suppose the graph of a linear function passes through the point $(x_1,y_1)$ and the slope of the line is $m.$ Since any other point $(x,f(x))$ on the graph of $f$ must satisfy the equation

this linear function can be expressed by writing

We call this equation the point-slope equation for that linear function.

Since every nonvertical line is the graph of a linear function, the points on a nonvertical line can be described using the slope-intercept or point-slope equations. However, a vertical line does not represent the graph of a function and cannot be expressed in either of these forms. Instead, a vertical line is described by the equation $x=k$ for some constant $k.$ Since neither the slope-intercept form nor the point-slope form allows for vertical lines, we use the notation

where $a,b$ are both not zero, to denote the standard form of a line.

Example 1.12

Finding the Slope and Equations of Lines

Consider the line passing through the points $(11,\mathrm{-4})$ and $(\mathrm{-4},5),$ as shown in Figure 1.17.

Figure 1.17 Finding the equation of a linear function with a graph that is a line between two given points.

1. Find the slope of the line.
2. Find an equation for this linear function in point-slope form.
3. Find an equation for this linear function in slope-intercept form.

Solution

1. The slope of the line is
   
   $$m=\frac{y_2-y_1}{x_2-x_1}=\frac{5-(\mathrm{-4})}{\mathrm{-4}-11}=-\frac{9}{15}=-\frac{3}{5}.$$
2. To find an equation for the linear function in point-slope form, use the slope $m=\mathrm{-3}\text{/}5$ and choose any point on the line. If we choose the point $(11,\mathrm{-4}),$ we get the equation
   
   $$f(x)+4=-\frac{3}{5}(x-11).$$
3. To find an equation for the linear function in slope-intercept form, solve the equation in part b. for $f(x).$ When we do this, we get the equation
   
   $$f(x)=-\frac{3}{5}x+\frac{13}{5}.$$

Polynomials

A linear function is a special type of a more general class of functions: polynomials. A polynomial function is any function that can be written in the form

for some integer $n\ge 0$ and constants $a_n,a_{n-1}\text{,…,}{a}_0,$ where $a_n\ne 0.$ In the case when $n=0,$ we allow for $a_0=0;$ if $a_0=0,$ the function $f(x)=0$ is called the zero function. The value $n$ is called the degree of the polynomial; the constant $a_n$ is called the leading coefficient. A linear function of the form $f\left(x\right)=mx+b$ is a polynomial of degree 1 if $m\ne 0$ and degree 0 if $m=0.$ A polynomial of degree 0 is also called a constant function. A polynomial function of degree 2 is called a quadratic function. In particular, a quadratic function has the form $f\left(x\right)=a{x}^2+bx+c,$ where $a\ne 0.$ A polynomial function of degree $3$ is called a cubic function.

Power Functions

Some polynomial functions are power functions. A power function is any function of the form $f(x)=a{x}^b,$ where $a$ and $b$ are any real numbers. The exponent in a power function can be any real number, but here we consider the case when the exponent is a positive integer. (We consider other cases later.) If the exponent is a positive integer, then $f(x)=a{x}^n$ is a polynomial. If $n$ is even, then $f(x)=a{x}^n$ is an even function because $f(\text{−}\mathit{\text{x}})=a{(\text{−}\mathit{\text{x}})}^n=a{x}^n$ if $n$ is even. If $n$ is odd, then $f(x)=a{x}^n$ is an odd function because $f(\text{−}\mathit{\text{x}})=a{(\text{−}\mathit{\text{x}})}^n=\text{−}a{x}^n$ if $n$ is odd (Figure 1.18).

Figure 1.18 (a) For any even integer $n,f(x)=a{x}^n$ is an even function. (b) For any odd integer $n,f(x)=a{x}^n$ is an odd function.

Behavior at Infinity

To determine the behavior of a function $f$ as the inputs approach infinity, we look at the values $f(x)$ as the inputs, $x,$ become larger. For some functions, the values of $f(x)$ approach a finite number. For example, for the function $f\left(x\right)=2+1\text{/}x,$ the values $1\text{/}x$ become closer and closer to zero for all values of $x$ as they get larger and larger. For this function, we say $\text{“}f(x)$ approaches two as $x$ goes to infinity,” and we write $f\left(x\right)\to 2$ as $x\to \infty .$ The line $y=2$ is a horizontal asymptote for the function $f\left(x\right)=2+1\text{/}x$ because the graph of the function gets closer to the line as $x$ gets larger.

For other functions, the values $f(x)$ may not approach a finite number but instead may become larger for all values of $x$ as they get larger. In that case, we say $\text{“}f(x)$ approaches infinity as $x$ approaches infinity,” and we write $f\left(x\right)\to \infty$ as $x\to \infty .$ For example, for the function $f\left(x\right)=3x^2,$ the outputs $f(x)$ become larger as the inputs $x$ get larger. We can conclude that the function $f(x)=3x^2$ approaches infinity as $x$ approaches infinity, and we write $3x^2\to \infty$ as $x\to \infty .$ The behavior as $x\to \text{−}\infty$ and the meaning of $f\left(x\right)\to \text{−}\infty$ as $x\to \infty$ or $x\to \text{−}\infty$ can be defined similarly. We can describe what happens to the values of $f(x)$ as $x\to \infty$ and as $x\to \text{−}\infty$ as the end behavior of the function.

To understand the end behavior for polynomial functions, we can focus on quadratic and cubic functions. The behavior for higher-degree polynomials can be analyzed similarly. Consider a quadratic function $f\left(x\right)=a{x}^2+bx+c.$ If $a>0,$ the values $f\left(x\right)\to \infty$ as $x\to \text{±}\infty .$ If $a<0,$ the values $f\left(x\right)\to \text{−∞}$ as $x\to \text{±}\infty .$ Since the graph of a quadratic function is a parabola, the parabola opens upward if $a>0;$ the parabola opens downward if $a<0.$ (See Figure 1.19(a).)

Now consider a cubic function $f\left(x\right)=a{x}^3+b{x}^2+cx+d.$ If $a>0,$ then $f\left(x\right)\to \infty$ as $x\to \infty$ and $f\left(x\right)\to \text{−∞}$ as $x\to \text{−∞}.$ If $a<0,$ then $f\left(x\right)\to \text{−∞}$ as $x\to \infty$ and $f\left(x\right)\to \infty$ as $x\to \text{−∞}.$ As we can see from both of these graphs, the leading term of the polynomial determines the end behavior. (See Figure 1.19(b).)

Figure 1.19 (a) For a quadratic function, if the leading coefficient $a>0,$ the parabola opens upward. If $a<0,$ the parabola opens downward. (b) For a cubic function $f,$ if the leading coefficient $a>0,$ the values $f\left(x\right)\to \infty$ as $x\to \infty$ and the values $f\left(x\right)\to \text{−∞}$ as $x\to \text{−∞}.$ If the leading coefficient $a<0,$ the opposite is true.

Mathematical Models

A large variety of real-world situations can be described using mathematical models. A mathematical model is a method of simulating real-life situations with mathematical equations. Physicists, engineers, economists, and other researchers develop models by combining observation with quantitative data to develop equations, functions, graphs, and other mathematical tools to describe the behavior of various systems accurately. Models are useful because they help predict future outcomes. Examples of mathematical models include the study of population dynamics, investigations of weather patterns, and predictions of product sales.

As an example, let’s consider a mathematical model that a company could use to describe its revenue for the sale of a particular item. The amount of revenue $R$ a company receives for the sale of $n$ items sold at a price of $p$ dollars per item is described by the equation $R=p·n.$ The company is interested in how the sales change as the price of the item changes. Suppose the data in Table 1.6 show the number of units a company sells as a function of the price per item.

$\mathit{\text{p}}$	$6$	$8$	$10$	$12$	$14$
$\mathit{\text{n}}$	$19.4$	$18.5$	$16.2$	$13.8$	$12.2$

Table 1.6 Number of Units Sold $n$ (in Thousands) as a Function of Price per Unit $p$ (in Dollars)

In Figure 1.20, we see the graph the number of units sold (in thousands) as a function of price (in dollars). We note from the shape of the graph that the number of units sold is likely a linear function of price per item, and the data can be closely approximated by the linear function $n=\mathrm{-1.04}p+26$ for $0\le p\le 25,$ where $n$ predicts the number of units sold in thousands. Using this linear function, the revenue (in thousands of dollars) can be estimated by the quadratic function

$$R\left(p\right)=p·\left(\mathrm{-1.04}p+26\right)=\mathrm{-1.04}{p}^2+26p$$

for $0\le p\le 25.$ In Example 1.15, we use this quadratic function to predict the amount of revenue the company receives depending on the price the company charges per item. Note that we cannot conclude definitively the actual number of units sold for values of $p,$ for which no data are collected. However, given the other data values and the graph shown, it seems reasonable that the number of units sold (in thousands) if the price charged is $p$ dollars may be close to the values predicted by the linear function $n=\mathrm{-1.04}p+26.$

Figure 1.20 The data collected for the number of items sold as a function of price is roughly linear. We use the linear function $n=\mathrm{-1.04}p+26$ to estimate this function.

Example 1.15

Maximizing Revenue

A company is interested in predicting the amount of revenue it will receive depending on the price it charges for a particular item. Using the data from Table 1.6, the company arrives at the following quadratic function to model revenue $R$ (in thousands of dollars) as a function of price per item $p\text{:}$

$$R\left(p\right)=p·\left(\mathrm{-1.04}p+26\right)=\mathrm{-1.04}{p}^2+26p$$

for $0\le p\le 25.$

1. Predict the revenue if the company sells the item at a price of $p=\text{\$}5$ and $p=\text{\$}17.$
2. Find the zeros of this function and interpret the meaning of the zeros.
3. Sketch a graph of $R.$
4. Use the graph to determine the value of $p$ that maximizes revenue. Find the maximum revenue.

Solution

1. Evaluating the revenue function at $p=5$ and $p=17,$ we can conclude that
   
   $$\begin{array}{}\\ R(5)=\mathrm{-1.04}{(5)}^2+26(5)=104,\text{so revenue}=\text{\$104,000;}\hfill \\ R(17)=\mathrm{-1.04}{(17)}^2+26(17)=141.44,\text{so revenue}=\text{\$141,440.}\hfill \end{array}$$
2. The zeros of this function can be found by solving the equation $\mathrm{-1.04}{p}^2+26p=0.$ When we factor the quadratic expression, we get $p\left(\mathrm{-1.04}p+26\right)=0.$ The solutions to this equation are given by $p=0,25.$ For these values of $p,$ the revenue is zero. When $p=\text{\$}0,$ the revenue is zero because the company is giving away its merchandise for free. When $p=\text{\$}25,$ the revenue is zero because the price is too high, and no one will buy any items.
3. Knowing the fact that the function is quadratic, we also know the graph is a parabola. Since the leading coefficient is negative, the parabola opens downward. One property of parabolas is that they are symmetric about the axis, so since the zeros are at $p=0$ and $p=25,$ the parabola must be symmetric about the line halfway between them, or $p=12.5.$
   
4. The function is a parabola with zeros at $p=0$ and $p=25,$ and it is symmetric about the line $p=12.5,$ so the maximum revenue occurs at a price of $p=\text{\$}12.50$ per item. At that price, the revenue is $R\left(p\right)=\mathrm{-1.04}{\left(12.5\right)}^2+26\left(12.5\right)=\text{\$}162,500.$

Algebraic Functions

By allowing for quotients and fractional powers in polynomial functions, we create a larger class of functions. An algebraic function is one that involves addition, subtraction, multiplication, division, rational powers, and roots. Two types of algebraic functions are rational functions and root functions.

Just as rational numbers are quotients of integers, rational functions are quotients of polynomials. In particular, a rational function is any function of the form $f\left(x\right)=p(x)\text{/}q\left(x\right),$ where $p(x)$ and $q(x)$ are polynomials. For example,

are rational functions. A root function is a power function of the form $f\left(x\right)=x^{1\text{/}n},$ where $n$ is a positive integer greater than one. For example, $f(x)=x^{1\text{/}2}=\sqrt{x}$ is the square-root function and $g\left(x\right)=x^{1\text{/}3}=\sqrt[3]{x}$ is the cube-root function. By allowing for compositions of root functions and rational functions, we can create other algebraic functions. For example, $f\left(x\right)=\sqrt{4-x^2}$ is an algebraic function.

The root functions $f\left(x\right)=x^{1\text{/}n}$ have defining characteristics depending on whether $n$ is odd or even. For all even integers $n\ge 2,$ the domain of $f\left(x\right)=x^{1\text{/}n}$ is the interval $[0,\infty ).$ For all odd integers $n\ge 1,$ the domain of $f\left(x\right)=x^{1\text{/}n}$ is the set of all real numbers. Since $x^{1\text{/}n}={-\left(\text{−}\mathit{\text{x}}\right)}^{1\text{/}n}$ for odd integers $n,f\left(x\right)=x^{1\text{/}n}$ is an odd function if $n$ is odd. See the graphs of root functions for different values of $n$ in Figure 1.21.

Figure 1.21 (a) If $n$ is even, the domain of $f\left(x\right)=\sqrt[n]{x}$ is $\left[0,\infty \right).$ (b) If $n$ is odd, the domain of $f\left(x\right)=\sqrt[n]{x}$ is $\left(\text{−∞},\infty \right)$ and the function $f\left(x\right)=\sqrt[n]{x}$ is an odd function.

Transcendental Functions

Thus far, we have discussed algebraic functions. Some functions, however, cannot be described by basic algebraic operations. These functions are known as transcendental functions because they are said to “transcend,” or go beyond, algebra. The most common transcendental functions are trigonometric, exponential, and logarithmic functions. A trigonometric function relates the ratios of two sides of a right triangle. They are $\sin x,\cos x,\tan x,\cot x,\sec x,\text{and}\csc x.$ (We discuss trigonometric functions later in the chapter.) An exponential function is a function of the form $f\left(x\right)=b^x,$ where the base $b>0,b\ne 1.$ A logarithmic function is a function of the form $f\left(x\right)={\log }_b(x)$ for some constant $b>0,b\ne 1,$ where ${\log }_b\left(x\right)=y$ if and only if $b^y=x.$ (We also discuss exponential and logarithmic functions later in the chapter.)