MA001: College Algebra

3a. perform algebraic operations on functions to construct and deconstruct composite functions

• Evaluate the sum, difference, product, and quotient of two functions. 
• Evaluate a composite function. Compare evaluating a composite function with simplifying the sum, difference, product, or quotient of two functions.

Finding the sum, difference, product, or quotient of two functions is equivalent to simplifying those computations with the functions, provided the input value is in the domain of both functions. That is, given two functions $f$ and $g$, the functions $f+g, f-g, f \cdot g$, and $\frac{f}{g}$ for $g \neq 0$ are evaluated by finding the output of each function for the indicated input and applying the required operation.

• $(f+g)(x)=f(x)+g(x)$
• $(f-g)(x)=f(x)-g(x)$
• $(f \cdot g)(x)=f(x) \cdot g(x)$
• $\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}$ for $g(x) \neq 0$

In contrast to the sum, difference, product, or quotient of two functions, a composite function is a function in which the output of one function becomes the input of the other function. The composite of functions $f$ and $g$, denoted as $f \circ g$ and read " $f$ of $g$ ", is defined as $(f \circ g)=f(g(x))$. That is, the output $g(x)$ becomes the input for function $f$.

Consider functions $f$ and $g$ defined as $f(x)=3 x+4$ and $g(x)=x^{2}$.

• $f(3)=13$
• $g(3)=9$
• $(f+g)(3)=f(3)+g(3)=13+9=22$
• $(f-g)(3)=f(3)-g(3)=13-9=4$
• $(f \cdot g)(3)=f(3) \cdot g(3)=13 \cdot 9=117$
• $\left(\frac{f}{g}\right)(3)=\frac{f(3)}{g(3)}=\frac{13}{9}$
• $(f \circ g)(3)=f(g(3))=f(9)=31$
• $(g \circ f)(3)=g(f(3))=g(13)=169$

Review the material in Creating and Evaluating Composite Functions.

3b. describe the properties of a composite function

• Determine whether the composition of two functions is a commutative operation.
• Explain how to find the domain of a composite function.
• How is function composition used to determine if two functions are inverses of each other?

In general, function composition is not commutative. For instance, given functions $f$ and $g$ defined as $f(x)=3 x+4$ and $g(x)=x^{2}$. Then $(f \circ g)(3)=f(g(3))=f(9)=31$ and $(g \circ f)(3)=g(f(3))=g(13)=169$. So, $(f \circ g)(x) \neq(g \circ f)(x)$.

Finding the domain of a composite function is a two-step process. To find the domain of $(f \circ g)(x)=f(g(x))$, it is important to know the domain of $f$ and $g$ and to determine those values in the range of $g$ that are in the domain of $f$. That is, the output of $g$ becomes the input or domain of $f$.

Suppose functions $h$ and $k$ are defined as $h(x)=\sqrt{x^{2}-4}$ and $k(x)=x^{2}$. For function $h$, the domain is $(-\infty,-2] \cup[2, \infty)$ and the range is $[0, \infty)$. The domain of function $k$ is all real numbers and the range is all nonnegative real numbers, that is, the domain is $(-\infty, \infty)$ and the range is $[0, \infty)$. Then, to find the domain of $h \circ k$, consider those values in the domain of function $k$ whose resulting output values are in the domain of function $h$; that is, of all real numbers in the domain of function $k$, only select those values whose output is in the domain of function $h$. Hence, the domain of $h \circ k$ is $(-\infty,-\sqrt{2}] \cup[\sqrt{2}, \infty)$. In contrast, the domain for $k \circ h$ will be those values in the domain of function $h$ whose output values are in the domain of function $k$. The only real numbers from the domain of function $k$ that are feasible in function $h$ are those in the domain of function $h$. So the domain of $k \circ h$ is $(-\infty,-2] \cup[2, \infty)$.

For two functions $p$ and $q$, if $p(q(x))=q(p(x))=x$ for all values of $x$ in the input of both $p$ and $q$, then the two functions $p$ and $q$ are inverses of each other. That is, inverse functions undo each other.

Review the material in Creating and Evaluating Composite Functions and Inverse Functions.

3c. identify basic graphical transformations of functions, including shifts, compressions, stretches, and reflections

• Describe the result of a vertical shift on the graph of a function. How is the equation $y=f(x)$ modified to indicate a vertical shift?
• Describe the result of a horizontal shift on the graph of a function. How is the equation $y=f(x)$ modified to indicate a horizontal shift?
• Describe the result of a stretch or compression on the graph of a function. How is the equation $y=f(x)$ modified to indicate a stretch or compression?
• Compare the result of a horizontal reflection with the result of a vertical reflection. How is the equation $y=f(x)$ modified to indicate either a vertical or horizontal reflection?

Functions can be transformed in a number of ways, including shifts left or right, shifts up or down, stretches or compressions, or reflections across either the x- or y-axis. Each type of transformation can be represented by an appropriate change in the definition of the function. 

One common transformation is a translation, that is, either a vertical shift (up or down) or a horizontal shift (left or right). A vertical shift is denoted by adding a constant to the function, modifying $y=f(x)$ to $y=f(x)+k$ with a positive value of $k$ indicating a shift up and a negative value of $k$ indicating a shift down. A horizontal shift is denoted by adjusting the input value of the function, modifying $y=f(x)$ to $y=f(x-h)$ with a positive value of $h$ indicating a shift right and a negative value of $h$ indicating a shift left. Thus, the graph of $y=$ $(x+5)^{2}+6$ represents a transformation of the graph of $y=x^{2}$ when shifted to the left 5 units and up 6 units. Notice, that to determine the horizontal shift, it is important to think of the equation as $y=[x-(-5)]^{2}+6$.

A second common transformation is a stretch or compression of a graph. A vertical stretch or compression of the graph of $y=f(x)$ is denoted by $y=a f(x)$, with the transformation being a stretch if $a > 1$ and a compression if $0 < a < 1$. Likewise, a horizontal stretch or compression is denoted by $y=f(b x)$ with the transformation being a stretch by a factor of $\frac{1}{b}$ if $0 < b < 1$ and a compression by a factor of $\frac{1}{b}$ if $b > 1$. Thus, the graph of $y=3 x^{2}$ represents a vertical stretch of the graph of $y=x^{2}$ by a factor of 3 ; the graph of $y=(5 x)^{2}$ represents a horizontal compression of the graph of $y=x^{2}$ by a factor of $\frac{1}{5}$.

A third common transformation is a reflection either vertically across the $x$-axis or horizontally across the $y$ axis. A vertical reflection across the $x$-axis is denoted by $y=-f(x)$; a horizontal reflection across the $y$-axis is denoted by $y=f(-x)$. Thus, the graph of $y=-x^{2}$ is a reflection of the graph of $y=x^{2}$ across the $x$-axis; the graph of $y=(-x)^{3}$ is a reflection of the graph of $y=x^{3}$ across the $y$-axis.

When multiple transformations are incorporated into the graph or equation for the function, it is important to interpret them in a particular order just as computations are completed in a particular order. For vertical shifts and stretches/compressions in the form $y=a f(x)+k$, complete the vertical stretch and then complete the vertical shift. For horizontal shifts and stretches/compressions in the form $y=a f(b(x-h))$, complete the horizontal stretch/compression and then the horizontal shift; if necessary, rewrite the function so that it is in this form.

Review the material in Graphing Functions Using Vertical and Horizontal Shifts.

3d. determine whether a function is even, odd, or neither

• Identify the characteristics of an even function and an odd function.
• What test can be used to determine if a function is even, odd, or neither?

Functions sometimes have symmetry. A function symmetric about the $y$-axis is an even function; a function symmetric about the origin is an odd function. If $y=f(x)=f(-x)$, then the function is an even function. In contrast, if $y=f(x)=-f(-x)$, then the function is an odd function. A function can be even, odd, or neither.

Consider functions $f, g$, and $h$ as shown below. Function $f$ defined as $f(x)=x^{4}$ is an even function; for every $x$, $f(x)=f(-x)$ and the graph is symmetric about the $y$-axis. Function $g$ defined as $g(x)=x^{5}$ is an odd function; for every $x, f(x)=-f(-x)$ and the graph is symmetric about the origin. Function $h$ defined as $h(x)=(x+1)^{2}-3$ is neither even nor odd; $h(x) \neq h(-x)$ and $h(x) \neq-h(-x)$ nor is the graph symmetric to either the $y$-axis or the origin.

Review the material in Graphing Functions Using Vertical and Horizontal Shifts.

3e. verify the inverse of a function

• Determine the conditions under which a function has an inverse which is a function.
• Explain how to verify if two functions are inverses of each other algebraically. 

An inverse function is a function whose input is the output of the original function and whose output is the input of the original function. The inverse of function $f$ is typically denoted $f^{-1}$, but this notation does not mean the reciprocal of $f$. If the ordered pair $(7, \, 10)$ is an element of the original function, then the ordered pair $(10,7)$ would be an element of the inverse function.

A function only has an inverse if the function is one-to-one so that each input value has just one output value and each output value has just one input value. If the function is not one-to-one, then there is at least one output value paired with two different input values. So, switching the input values and output values means the new input value would be paired with two different output values, so the new set of ordered pairs is not a function.

Two functions are inverses of each other if $\left(f \circ f^{-1}\right)(x)=x$ for all $x$ in the domain of $f^{-1}$ and $\left(f^{-1} \circ f\right)(x)=x$ for all $x$ in the domain of $f$. For example, let $f(x)=4 x+3$ and $g(x)=4 x-3$. Because $(f \circ g)(5)=f(g(5))=$ $f(17)=71$, the two functions are not inverses of each other. In contrast, if $h(x)=\frac{x-3}{4}$, then $(f \circ h)(5)=$ $f(h(5))=f\left(\frac{1}{2}\right)=5$ and $(h \circ f)(5)=h(f(5))=h(23)=\frac{23-3}{4}=5$. Thus, functions $f$ and $h$ are inverses of each other.

Review material in Inverse Functions.

3f. define the inverse of a function and its domain and range using algebraic operations

• Explain how to determine the domain and range of an inverse function.
• Identify how the inverse of a function can be determined algebraically.

In general, the domain of a function represents the range of its inverse; likewise, the range of the original function is the domain of the inverse. If the original function is not one-to-one, then a restriction must be placed on the function to identify an appropriate inverse function; the restriction can be any interval over which the restricted function is one-to-one.

If a one-to-one function is represented by a table, then find the inverse of the function by reversing the input and output values. If a one-to-one function is represented by an equation $y=f(x)$, find the inverse by reversing the input and output values and solving for a new value of $y$. For example, let a function $f$ be defined as $y=f(x)=7 x+8$. To find the inverse, reverse $x$ and $y$ and then solve for $y: x=7 y+8$ so $y=\frac{x-8}{7}$. So, the inverse function $f^{-1}$ is defined as $f^{-1}(x)=\frac{x-8}{7}$.

Review material in Inverse Functions.

3g. given a one-to-one function construct the graph of its inverse

Specify the equation of the identity line.

Describe how the identity line can be used to graph the inverse of a function.

The identity line has equation $y=x$. If the point $(a, b)$ is on the graph of a function, then the point $(b, a)$ is on the graph of its inverse. These two points are reflections of each other across the identity line. So, the identity line is helpful in graphing a function and its inverse. If a function is one-to-one so that it has an inverse, the graph of its inverse is the reflection of the graph of the function across the identity line $y=x$.

Consider the graph of a function and its inverse on the coordinate grid below. Observe that the ordered pairs $(3,0)$ and $(12,3)$ are both on the graph of function $f$; thus the ordered pairs $(0,3)$ and $(3,12)$ must be on the graph of the inverse function $f^{-1}$.

Review material in Inverse Functions.

Unit 3 Vocabulary

This vocabulary list includes terms you will need to know to successfully complete the final exam.

• composite function
• compression of a graph
• even function
• horizontal shift
• horizontal reflection
• identity line
• inverse function
• odd function
• reflection
• stretch of a graph
• translation
• vertical reflection
• vertical shift