Functions and Function Notation

A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

Determining whether a relation represents a function

A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.

The domain is $\{1,2,3,4,5\}$. The range is $\{2,4,6,8,10\}$.

Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter $x$. Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter $y$.

A function $f$ is a relation that assigns a single value in the range to each value in the domain. In other words, no $x$-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, $\{1,2,3,4,5\}$, is paired with exactly one element in the range, $\{2,4,6,8,10\}$.

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

Notice that each element in the domain, $\{\text{even},\text{odd}\}$ is not paired with exactly one element in the range, $\{1,2,3,4,5\}$. For example, the term “odd” corresponds to three values from the range, $\{1,3,5\}$, and the term “even” corresponds to two values from the range, $\{2,4\}$. This violates the definition of a function, so this relation is not a function.

The three diagrams below compare relations that are functions and not functions.

Diagram (a) is a function because each input is associated with a single output. Note that inputs $q$ and $r$ both give output $n$. Diagram (b) is also a function. In this case, each input is associated with a single output. Diagram (c) is not a function because input $q$ is associated with two different outputs.

Example. The coffee shop menu below consists of items and their prices.

(a) Is price a function of the item?

(b) Is the item a function of the price?

Solution. (a) Let’s begin by considering the input as the items on the menu. The output values are then the prices.

Each item on the menu has only one price, so the price is a function of the item.

(b) Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it.

Therefore, the item is not a function of price.

Example. In a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? The table below shows a possible rule for assigning grade points.

Solution. For any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specific grade point average.

In the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average.

The table below lists the five greatest baseball players of all time in order of rank.

Using function notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.

To represent “height is a function of age,” we start by identifying the descriptive variables $h$ for height and $a$ for age. The letters $f$, $g$, and $h$ are often used to represent functions just as we use $x$, $y$, and $z$ to represent numbers and $A$, $B$, and $C$ to represent sets.

Remember, we can use any letter to name the function; the notation $h(a)$ shows us that $h$ depends on $a$. The value $a$ must be put into the function $h$ to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.

We can also give an algebraic expression as the input to a function. For example $f(a+b)$ means “first add $a$ and $b$, and the result is the input for the function $f$.” The operations must be performed in this order to obtain the correct result.

Example. Use function notation to represent a function whose input is the name of a month and output is the number of days in that month. Assume that the domain does not include leap years.

Solution. The number of days in a month is a function of the name of the month, so if we name the function $f$, we write $\text{days}=f(\text{month})$ or $d=f(m)$. The name of the month is the input to a “rule” that associates a specific number (the output) with each input.

For example, $f(\text{March})=31$, because March has 31 days. The notation $d=f(m)$ reminds us that the number of days, $d$ (the output), is dependent on the name of the month, $m$ (the input).

Analysis. Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.

Example. A function $N=f(y)$ gives the number of police officers, $N$, in a town in year $y$. What does $f(2005)=300$ represent?

Solution. When we read $f(2005)=300$, we see that the input year is 2005. The value for the output, the number of police officers $(N)$, is 300. Remember, $N=f(y)$. The statement $f(2005)=300$ tells us that in the year 2005 there were 300 police officers in the town.

Representing functions using tables

A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.

The table below lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function $f$ where $D=f(m)$ identifies months by an integer rather than by name.

The next table defines a function $Q=g(n)$. Remember, this notation tells us that $g$ is the name of the function that takes the input $n$ and gives the output $Q$.

The third table displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.

Example. Which table, A, B, or C, represents a function (if any)?

Table A:

Table B:

Table C:

Solution. Tables A and B define functions. In both, each input value corresponds to exactly one output value. Table C does not define a function because the input value of 5 corresponds to two different output values.

When a table represents a function, corresponding input and output values can also be specified using function notation. The function represented by table A can be represented by writing

Similarly, the statements

represent the function in table B. Table C cannot be expressed in a similar way because it does not represent a function.

Finding input and output values of a function

When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.

When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function’s formula and solve for the input. Solving can produce more than one solution because different input values can produce the same output value.

Evaluation of functions in algebraic forms

When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function $f(x)=5-3x^2$ can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.

Example. Evaluate $f(x)=x^2+3x-4$ at:

(a) $2$   (b) $a$   (c) $a+h$   (d) Now evaluate $\tfrac{f(a+h)-f(a)}{h}$

Solution. Replace the $x$ in the function with each specified value.

(a) Because the input value is a number, 2, we can use simple algebra to simplify.

(b) In this case, the input value is a letter so we cannot simplify the answer any further.

(c) With an input value of $a+h$, we must use the distributive property.

(d) In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that

and we know that

Now we combine the results and simplify.

Example. Given the function $h(p)=p^2+2p$, evaluate $h(4)$.

Solution. To evaluate $h(4)$, we substitute the value 4 for the input variable $p$ in the given function.

Therefore, for an input of 4, we have an output of 24.

Example. Given the function $h(p)=p^2+2p$, solve for $h(p)=3$.

Solution.

If $(p+3)(p-1)=0$, either $(p+3)=0$ or $(p-1)=0$ (or both of them equal 0). We will set each factor equal to 0 and solve for $p$ in each case.

This gives us two solutions. The output $h(p)=3$ when the input is either $p=1$ or $p=-3$. We can also verify by graphing. The graph verifies that $h(1)=h(-3)=3$ and $h(4)=24$.

Evaluating functions expressed in formulas

Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation $2n+6p=12$ expresses a functional relationship between $n$ and $p$. We can rewrite it to decide if $p$ is a function of $n$.

Example. Express the relationship $2n+6p=12$ as a function $p=f(n)$, if possible.

Solution. To express the relationship in this form, we need to be able to write the relationship where $p$ is a function of $n$, which means writing it as $p=[\text{expression involving }n]$.

Therefore, $p$ as a function of $n$ is written as

Analysis. It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.

Example. Does the equation $x^2+y^2=1$ represent a function with $x$ as input and $y$ as output? If so, express the relationship as a function $y=f(x)$.

Solution. First we subtract $x^2$ from both sides.

We now try to solve for $y$ in this equation.

We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function $y=f(x)$.

Evaluating a function given in tabular form

As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy’s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.

The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table.

At times, evaluating a function in table form may be more useful than using equations. Here let us call the function $P$. The domain of the function is the type of pet and the range is a real number representing the number of hours the pet’s memory span lasts. We can evaluate the function $P$ at the input value of “goldfish.” We would write $P(\text{goldfish})=2160$. Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function $P$ seems ideally suited to this function, more so than writing it in paragraph or function form.

Example. Using the table below, (a) evaluate $g(3)$, and (b) solve $g(n)=6$.

Solution. (a) Evaluating $g(3)$ means determining the output value of the function $g$ for the input value of $n=3$. The table output value corresponding to $n=3$ is 7, so $g(3)=7$.

(b) Solving $g(n)=6$ means identifying the input values, $n$, that produce an output value of 6. The table shows two solutions: $2$ and $4$. When we input 2 into the function $g$, our output is 6. When we input 4 into the function $g$, our output is also 6.

Finding function values from a graph

Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).

Example. Given the graph below, (a) evaluate $f(2)$, and (b) solve $f(x)=4$.

Solution. (a) To evaluate $f(2)$, locate the point on the curve where $x=2$, then read the $y$-coordinate of that point. The point has coordinates $(2,1)$, so $f(2)=1$.

(b) To solve $f(x)=4$, we find the output value $4$ on the vertical axis. Moving horizontally along the line $y=4$, we locate two points of the curve with output value $4$: $(-1,4)$ and $(3,4)$. These points represent the two solutions to $f(x)=4$: $-1$ or $3$. This means $f(-1)=4$ and $f(3)=4$, or when the input is $-1$ or $3$, the output is $4$.

Determining whether a function is one-to-one

Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in the figure at the beginning of this chapter, the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.

However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed below.

This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.

To visualize this concept, let’s look again at the two simple functions sketched in diagrams (a) and (b) at the start of this section. The function in diagram (a) shows a relationship that is not a one-to-one function because inputs $q$ and $r$ both give output $n$. The function in diagram (b) shows a relationship that is a one-to-one function because each input is associated with a single output.

Example. Is the area of a circle a function of its radius? If yes, is the function one-to-one?

Solution. A circle of radius $r$ has a unique area measure given by $A=\pi r^2$, so for any input, $r$, there is only one output, $A$. The area is a function of radius $r$.

If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure $A$ is given by the formula $A=\pi r^2$. Because areas and radii are positive numbers, there is exactly one solution: $\sqrt{\tfrac{A}{\pi}}$. So the area of a circle is a one-to-one function of the circle’s radius.

Using the vertical line test

As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.

The most common graphs name the input value $x$ and the output value $y$, and we say $y$ is a function of $x$, or $y=f(x)$ when the function is named $f$. The graph of the function is the set of all points $(x,y)$ in the plane that satisfies the equation $y=f(x)$. If the function is defined for only a few input values, then the graph of the function is only a few points, where the $x$-coordinate of each point is an input value and the $y$-coordinate of each point is the corresponding output value. For example, the black dots on the graph below tell us that $f(0)=2$ and $f(6)=1$. However, the set of all points $(x,y)$ satisfying $y=f(x)$ is a curve. The curve shown includes $(0,2)$ and $(6,1)$ because the curve passes through those points.

The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value.

The first graph is a function: every vertical line meets it exactly once. The second and third are not functions, because a vertical line meets each of them at two points.

Example. Which of the graphs below represent(s) a function $y=f(x)$?

Solution. If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b). From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most $x$-values, a vertical line would intersect the graph at more than one point, as shown below.

Using the horizontal line test

Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.

Example. Consider the two functions graphed earlier in parts (a) and (b) of the vertical line test example. Are either of the functions one-to-one?

Solution. The function in part (a) is not one-to-one. The horizontal line shown below intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points).

The function in part (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.

Identifying basic toolkit functions

In this text, we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use $x$ as the input variable and $y=f(x)$ as the output variable.

We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties.

Constant — $f(x)=c$, where $c$ is a constant

Identity — $f(x)=x$

Absolute value — $f(x)=\lvert x\rvert$

Quadratic — $f(x)=x^2$

Cubic — $f(x)=x^3$

Reciprocal — $f(x)=\tfrac{1}{x}$

Reciprocal squared — $f(x)=\tfrac{1}{x^2}$

Square root — $f(x)=\sqrt{x}$

Cube root — $f(x)=\sqrt[3]{x}$

Key equations

Key concepts

• A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output.
• Function notation is a shorthand method for relating the input to the output in the form $y=f(x)$.
• In tabular form, a function can be represented by rows or columns that relate to input and output values.
• To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value.
• To solve for a specific function value, we determine the input values that yield the specific output value.
• An algebraic form of a function can be written from an equation.
• Input and output values of a function can be identified from a table.
• Relating input values to output values on a graph is another way to evaluate a function.
• A function is one-to-one if each output value corresponds to only one input value.
• A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.
• The graph of a one-to-one function passes the horizontal line test.

Key terms

relation — a set of ordered pairs. domain — the set of all input values of a relation. range — the set of all output values of a relation. independent variable — an input value of a function. dependent variable — an output value of a function. function — a relation in which each input value yields a unique output value. function notation — the notation $y=f(x)$, defining a function named $f$ with input $x$ and output $y$. one-to-one function — a function for which each value of the output is associated with a unique input value. vertical line test — a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once. horizontal line test — a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once.

This section is adapted from Precalculus 2e, Section 1.1: Functions and Function Notation by Jay Abramson and OpenStax, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated every figure as an accessible inline SVG, including the input–output mapping diagrams, the coffee shop menu mappings, and the function-machine diagram; presented the menu itself and the toolkit function library as Markdown tables and headed figure groups rather than images; omitted the media links and end-of-section exercises; converted the practice problems (“Try Its”) into interactive exercises with instant feedback; and presented the write-a-function-notation practice item as multiple choice so it can be graded in the browser.