Basic Classes of Functions

We have studied the general characteristics of functions, so now let’s examine some specific classes of functions. We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define general algebraic functions and distinguish them from the transcendental functions we examine later in this chapter. We finish the section with examples of piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form.

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 [link], 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.

As suggested by [link], 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 [link], we see this ratio is independent of the points chosen.

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.

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 ([link]).

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 [link](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 [link](b).)

Zeros of Polynomial Functions

Another characteristic of the graph of a polynomial function is where it intersects the $x$

-axis. To determine where a function $f$

intersects the $x$

-axis, we need to solve the equation $f(x)=0$

for .n the case of the linear function $f(x)=mx+b,$

the $x$

-intercept is given by solving the equation $mx+b=0.$

In this case, we see that the $x$

-intercept is given by $(\text{−}\mathit{\text{b}}\text{/}m,0).$

In the case of a quadratic function, finding the $x$

-intercept(s) requires finding the zeros of a quadratic equation: $a{x}^2+bx+c=0.$

In some cases, it is easy to factor the polynomial $a{x}^2+bx+c$

to find the zeros. If not, we make use of the quadratic formula.

In the case of higher-degree polynomials, it may be more complicated to determine where the graph intersects the $x$

-axis. In some instances, it is possible to find the $x$

-intercepts by factoring the polynomial to find its zeros. In other cases, it is impossible to calculate the exact values of the $x$

-intercepts. However, as we see later in the text, in cases such as this, we can use analytical tools to approximate (to a very high degree) where the $x$

-intercepts are located. Here we focus on the graphs of polynomials for which we can calculate their zeros explicitly.

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 [link] show the number of units a company sells as a function of the price per item.

In [link], 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

for $0\le p\le 25.$

In [link], 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 $n=\mathrm{-1.04}p+26.$

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 [link].

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.)

Piecewise-Defined Functions

Sometimes a function is defined by different formulas on different parts of its domain. A function with this property is known as a piecewise-defined function. The absolute value function is an example of a piecewise-defined function because the formula changes with the sign of $x\text{:}$

Other piecewise-defined functions may be represented by completely different formulas, depending on the part of the domain in which a point falls. To graph a piecewise-defined function, we graph each part of the function in its respective domain, on the same coordinate system. If the formula for a function is different for $x<a$

and $x>a,$

we need to pay special attention to what happens at $x=a$

when we graph the function. Sometimes the graph needs to include an open or closed circle to indicate the value of the function at $x=a.$

We examine this in the next example.

Transformations of Functions

We have seen several cases in which we have added, subtracted, or multiplied constants to form variations of simple functions. In the previous example, for instance, we subtracted 2 from the argument of the function $y=x^2$

to get the function $f\left(x\right)={\left(x-2\right)}^2.$

This subtraction represents a shift of the function $y=x^2$

two units to the right. A shift, horizontally or vertically, is a type of transformation of a function. Other transformations include horizontal and vertical scalings, and reflections about the axes.

A vertical shift of a function occurs if we add or subtract the same constant to each output $y.$

For $c>0,$

the graph of $f\left(x\right)+c$

is a shift of the graph of $f(x)$

up $c$

units, whereas the graph of $f\left(x\right)-c$

is a shift of the graph of $f\left(x\right)$

down $c$

units. For example, the graph of the function $f(x)=x^3+4$

is the graph of $y=x^3$

shifted up $4$

units; the graph of the function $f(x)=x^3-4$

is the graph of $y=x^3$

shifted down $4$

units ([link]).

A horizontal shift of a function occurs if we add or subtract the same constant to each input $x.$

For $c>0,$

the graph of $f(x+c)$

is a shift of the graph of $f(x)$

to the left $c$

units; the graph of $f(x-c)$

is a shift of the graph of $f(x)$

to the right $c$

units. Why does the graph shift left when adding a constant and shift right when subtracting a constant? To answer this question, let’s look at an example.

Consider the function $f\left(x\right)=\backslash |x+3\backslash |$

and evaluate this function at $x-3.$

Since $f\left(x-3\right)=\backslash |x\backslash |$

and $x-3<x,$

the graph of $f\left(x\right)=\backslash |x+3\backslash |$

is the graph of $y=\backslash |x\backslash |$

shifted left 3 units. Similarly, the graph of $f\left(x\right)=\backslash |x-3\backslash |$

is the graph of $y=\backslash |x\backslash |$

shifted right $3$

units ([link]).

A vertical scaling of a graph occurs if we multiply all outputs $y$

of a function by the same positive constant. For $c>0,$

the graph of the function $cf(x)$

is the graph of $f(x)$

scaled vertically by a factor of $c.$

If $c>1,$

the values of the outputs for the function $cf(x)$

are larger than the values of the outputs for the function $f(x);$

therefore, the graph has been stretched vertically. If $0<c<1,$

then the outputs of the function $cf(x)$

are smaller, so the graph has been compressed. For example, the graph of the function $f\left(x\right)=3x^2$

is the graph of $y=x^2$

stretched vertically by a factor of 3, whereas the graph of $f\left(x\right)=x^2\text{/}3$

is the graph of $y=x^2$

compressed vertically by a factor of $3$

([link]).

The horizontal scaling of a function occurs if we multiply the inputs $x$

by the same positive constant. For $c>0,$

the graph of the function $f(cx)$

is the graph of $f(x)$

scaled horizontally by a factor of $c.$

If $c>1,$

the graph of $f(cx)$

is the graph of $f(x)$

compressed horizontally. If $0<c<1,$

the graph of $f\left(cx\right)$

is the graph of $f\left(x\right)$

stretched horizontally. For example, consider the function $f\left(x\right)=\sqrt{2x}$

and evaluate $f$

at $x\text{/}2.$

Since $f(x\text{/}2)=\sqrt{x},$

the graph of $f\left(x\right)=\sqrt{2x}$

is the graph of $y=\sqrt{x}$

compressed horizontally. The graph of $y=\sqrt{x\text{/}2}$

is a horizontal stretch of the graph of $y=\sqrt{x}$

([link]).

We have explored what happens to the graph of a function $f$

when we multiply $f$

by a constant $c>0$

to get a new function $cf(x).$

We have also discussed what happens to the graph of a function $f$

when we multiply the independent variable $x$

by $c>0$

to get a new function $f(cx).$

However, we have not addressed what happens to the graph of the function if the constant $c$

is negative. If we have a constant $c<0,$

we can write c as a positive number multiplied by $\mathrm{-1};$

but, what kind of transformation do we get when we multiply the function or its argument by $\mathrm{-1}?$

When we multiply all the outputs by $\mathrm{-1},$

we get a reflection about the $x$

-axis. When we multiply all inputs by $\mathrm{-1},$

we get a reflection about the $y$

-axis. For example, the graph of $f\left(x\right)=\text{−}(x^3+1)$

is the graph of $y=(x^3+1)$

reflected about the $x$

-axis. The graph of $f\left(x\right)={\left(\text{−}x\right)}^3+1$

is the graph of $y=x^3+1$

reflected about the $y$

-axis ([link]).

If the graph of a function consists of more than one transformation of another graph, it is important to transform the graph in the correct order. Given a function $f(x),$

the graph of the related function $y=cf\left(a\left(x+b\right)\right)+d$

can be obtained from the graph of $y=f(x)$

by performing the transformations in the following order.

1. Horizontal shift of the graph of $y=f(x).$

   If

   $b>0,$

   shift left. If

   $b<0,$

   shift right.

2. Horizontal scaling of the graph of $y=f(x+b)$

   by a factor of

   $\backslash |a\backslash |.$

   If

   $a<0,$

   reflect the graph about the

   $y$

   -axis.

3. Vertical scaling of the graph of $y=f(a\left(x+b\right))$

   by a factor of

   $\backslash |c\backslash |.$

   If

   $c<0,$

   reflect the graph about the

   $x$

   -axis.

4. Vertical shift of the graph of $y=cf(a\left(x+b\right)).$

   If

   $d>0,$

   shift up. If

   $d<0,$

   shift down.

We can summarize the different transformations and their related effects on the graph of a function in the following table.

Key Concepts

• The power function $f\left(x\right)=x^n$

  is an even function if

  $n$

  is even and

  $n\ne 0,$

  and it is an odd function if

  $n$

  is odd.

• The root function $f\left(x\right)=x^{1\text{/}n}$

  has the domain

  $[0,\infty )$

  if

  $n$

  is even and the domain

  $(\text{−∞},\infty )$

  if

  $n$

  is odd. If

  $n$

  is odd, then

  $f\left(x\right)=x^{1\text{/}n}$

  is an odd function.

• The domain of the rational function $f\left(x\right)=p\left(x\right)\text{/}q\left(x\right),$

  where

  $p\left(x\right)$

  and

  $q\left(x\right)$

  are polynomial functions, is the set of

  $x$

  such that

  $q\left(x\right)\ne 0.$
• Functions that involve the basic operations of addition, subtraction, multiplication, division, and powers are algebraic functions. All other functions are transcendental. Trigonometric, exponential, and logarithmic functions are examples of transcendental functions.
• A polynomial function $f$

  with degree

  $n\ge 1$

  satisfies

  $f\left(x\right)\to \text{±}\infty$

  as

  $x\to \text{±}\infty .$

  The sign of the output as

  $x\to \infty$

  depends on the sign of the leading coefficient only and on whether

  $n$

  is even or odd.

• Vertical and horizontal shifts, vertical and horizontal scalings, and reflections about the $x$

  - and

  $y$

  -axes are examples of transformations of functions.

Key Equations

• Point-slope equation of a line

  ---

  $y-y_1=m\left(x-x_1\right)$
• Slope-intercept form of a line

  ---

  $y=mx+b$
• Standard form of a line

  ---

  $ax+by=c$
• Polynomial function

  ---

  $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\text{⋯}+a_{1}x+a_0$

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