MA001 Study Guide

Unit 2: Introduction to Functions

2a. summarize the properties of a function, including input and output values, domain, and range using words, function notation, and tables

Distinguish between a relation and a function.
Identify the difference between input and output values. Relate input and output values to independent and dependent variables and domain and range.
Be able to interpret function notation.
Determine when a function is one-to-one.

Any set of ordered pairs represents a relation. The first element in the ordered pair represents the input value or independent variable; the second element in the ordered pair represents the output value or dependent variable. The set of all input or independent values describes the domain of the relation; the set of all output or dependent values represents the range of the relation.

A relation is a function if each input value is associated with a single output value. Compare the following two sets of ordered pairs.

Both Sets $A$ and $B$ are relations. In Set $A$ , each input value is associated with two different output values, so Set $A$ is not a function. Its domain is $\{1,2,3\}$ and its range is $\{2, \, 4, \, 6, \, 8,\, 12\}$ . In Set $B$ , each input value is associated with just one output value so Set B is a function. Its domain is $\{1,2,3\}$ and its range is $\{2, \, 4, \,6\}$ .

Functions are typically represented using equations or function notation. In Set B above, each output value is twice the input value. So, if $x$ represents the input value and $y$ represents the output value, the function could be described as $y=2 x$ . If $f$ names the function, then the function could also be described as $f(x)=2 x$ , read " $f$ of $x$ is twice $x$ ". Notice that the function is described by indicating the input value for the function (in this case $x$ ) and then the rule applied to that input value (in this case $2$ times $x$ ).

A function is a one-to-one function if each output value is associated with a single input value. For instance, $\{(1, \, 2),(2, \, 2),(3, \, 2)\}$ is a function because each input value is associated with just one output value; however, it is not one-to-one because the output value 2 is associated with more than one input value. In contrast, $\{(1, \, 2)$ , $(2, \, 4),(3, \, 6)\}$ is a function because each input value is associated with just one output value; it is a one-to-one function because each output value is associated with just one input value.

Review the material in Defining and Writing Functions.

2b. evaluate a function given an equation, table, or words

Be able to evaluate a function represented in function notation.
Compare evaluating a function using function notation to evaluating a function represented by a table of values.

To evaluate a function, identify the input or independent variable and then apply the function rule to that variable. For instance, given the function $f$ defined as $f(x)=3 x^{2}+5 x^{3}$ . The input variable is the variable inside the parentheses, namely $x$ , and the rule for the function is $3 x^{2}+5 x^{3}$ . So, to evaluate $f(4)$ , substitute 4 for $x$ and simplify: $3 \cdot 4^{2}+5 \cdot 4^{3}=3 \cdot 16+5 \cdot 64=368$ .

If a function is represented by a table, first identify the input values and the output values in the table. If the input value is actually listed in the table, then match the input value with its associated output value. If the input value is not actually listed in the table, then attempt to find a rule to describe how the input values are associated with the output values and use this rule to evaluate the function for a non-listed input value.

Review the material in Defining and Writing Functions.

2c. identify whether the graph of a relation represents a function

Explain how to use the vertical line test to determine if the graph of a relation represents a function.
Explain how to use the horizontal line test to determine if a function is one-to-one.

In looking at a graph of a relation to determine whether it represents a function, it is important to consider whether each first element of an ordered pair is associated with a single second element of the ordered pair. If so, the relation is a function; if not, the relation is not a function. One quick way to check whether a graph represents a function is to use the vertical line test. If a vertical line intersects a graph in more than one point, then the graph does NOT represent a function because the input value associated with the vertical line is paired with more than one output value.

Once a graph is determined to represent a function, the horizontal line test can be applied to determine if the function is one-to-one. If the horizontal line intersects the graph in more than one point, then the graph does not represent a one-to-one function because the output value associated with the horizontal line is paired with more than one input value.

For instance, consider the graph of $y=f(x)$ shown on the coordinate grid below. The vertical line in red intersects the graph in only one point, even if it were to move across the coordinate plane from left to right; so the graph represents a function. However, the horizontal line in green intersects the graph in more than one point, so the function is not one-to-one.

Review the material in Defining and Writing Functions.

2d. use set builder, inequality, and interval notation to express the domain and range of a function defined by an equation, table, graph, or set

Identify the domain and range of a function expressed in a table or set of ordered pairs.
Identify the domain and range of a function given an equation. How are restrictions on the domain or range determined?
Identify the domain and range of a function represented by a graph.

If a function is expressed in a table or as a set of ordered pairs, then the domain will be the set of input values represented in the table or the set of first elements of the ordered pairs in the set. The range will be the set of output values represented in the table or the set of second elements of the ordered pairs in the set.

When identifying the domain and range of a function from an equation, it is important to consider whether there are any restrictions on the possible input values and the resulting output values. Such restrictions occur if particular input values would make the equation undefined or if particular output values can never be obtained.

For example, if $f(x)=4 x+5$ , the graph is a line; $x$ can represent any real number and the resulting $f(x)$ value can also be any real number. In contrast, consider $g(x)=x^{2}-3$ . In this case, although the input $x$ can be any real number, the output value can never be less than $-3$ . So the domain would be the set of all real numbers but the range would be $\{y: y \geq-3\}$ or $[-3, \infty)$ when represented in set-builder or interval notation, respectively. If $h(x)=\sqrt{x+4}$ , then the domain will be $x \geq-4$ because $x+4 \geq 0$ in order for the square root to be feasible; the domain is $y \geq 0$ because the value of the square root is always non-negative.

Similar processes are used to identify the domain and range of a graph identified on a coordinate plane. Here, one needs to consider whether endpoints on the graph suggest that the function is defined only over particular values or whether both the independent and dependent variables can extend to $\pm \infty$ . Compare the two graphs below. In $y=g(x)$ , the domain is $\{x:-4$ and the range is $\{y: 0 \leq y \leq 4\}$ . The open circle at $(-4, \, 2)$ indicates that $x=-4$ is not part of the domain; although $y=2$ would not be part of the range based on this point, $y=2$ is part of the domain from the point $(0, \, 2)$ . However, in $y=h(x)$ , the graph continues in both directions to infinity; so, the domain and range are both $(-\infty, \, \infty)$ .

Review the material in Finding Domain and Range from Graphs.

2e. construct the graph of a piecewise-defined function

What is a piecewise-defined function?
How do the domain and range help to graph a piecewise-defined function?

A piecewise-defined function is a function in which the output is defined differently for various values of the domain. The overall domain of the function is the union of the domain of each piece. Similarly, the overall range of the function is the union of the range of each piece. Compare the graphs of the two functions below.

For function $p$ the domain is the set of all real numbers; the range is $[-1, \infty)$ . For function $k$ , the domain is $(-\infty, \, 0] \cup[3, \, \infty)$ ; the range is $\{-2, \, 1\}$ . Notice that for function $k$ , there are some values of $x$ that are not part of the domain and the range consists of just two values.

Review the material in Finding Domain and Range from Graphs.

2f. calculate the average rate of change of a function given a table, graph, or equation

Explain how to determine the average rate of change of a function over a specified interval.
Compare the average rate of change of a linear function with the average rate of change of a nonlinear function.

The average rate of change of a function describes how the output values of a function change over a particular interval of input values. In general, if $y=f(x)$ , then

average rate of change of $f=\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{2}}$

As an example, consider a function $g$ defined as $g(x)=x^{2}-7 x$ . To find the average rate of change of $g$ over the interval $[-4,3]$ , first find $g(-4)=44$ and $g(3)=-12$ . Then the average rate of change for $g$ in this interval is $\frac{-12-44}{3-(-4)}=\frac{-56}{7}=-8$ . If the function is represented by a graph or via a table, the average rate of change is found in the same way, by first identifying the output values from the graph or table and then simplifying the fraction related to the average rate of change.

To review, see Calculate the Rate of Change of a Function.

2g. identify the characteristics of a function given its graph, including behavior over an interval and local and absolute extrema

Identify intervals over which a function is increasing, decreasing, or constant.
Determine local minimum or local maximum values of a function over a specific interval.
Compare local minimum/maximum values with absolute minimum/maximum values.

It is often important to determine how a function behaves over a particular interval. If the output values increase as the input values increase, then the function is an increasing function; that is, function $f$ is increasing if $f\left(x_{2}\right) > f\left(x_{1}\right)$ when $x_{2} > x_{1}$ . Likewise, if the output values decrease as the input values increase, then the function is a decreasing function; that is, function $f$ is decreasing if $f\left(x_{2}\right)$ when $x_{2} > x_{1}$ . If the output values do not change as the input values increase, then the function is a constant function; that is, function $f$ is constant if $f\left(x_{2}\right)=f\left(x_{1}\right)$ when $x_{2} > x_{1}$ .

Local minimum or local maximum values occur where the function has a low value or high value, respectively, within a specific interval. If that low value or high value is the lowest value or highest value over the entire domain of the function, then it is an absolute minimum or absolute maximum value of the function, respectively.

Consider the function shown on the coordinate graph below. The function is decreasing for $(-\infty,-3)$ and $(-1,1)$ and increasing for $(-3,-1)$ and $(1, \infty)$ . The function has a local maximum value of 10 at $x=-1$ . The function has an absolute minimum value of $-6$ at both $x=-3$ and $x=1$ .