Algebra gives us a way to explore and describe relationships. Imagine tossing a ball straight up in the air and watching it rise to reach its highest point before dropping back down into your hands. As time passes, the height of the ball changes. There is a relationship between the amount of time that has elapsed since the toss and the height of the ball. In mathematics, a correspondence between variables that change together (such as time and height) is called a relation. Some, but not all, relations can also be described as functions.

There are many kinds of relations. Relations are simply correspondences between sets of values or information. Think about members of your family and their ages. The pairing of each member of your family and their age is a relation. Each family member can be paired with an age in the set of ages of your family members. Another example of a relation is the pairing of a state with its United States’ senators. Each state can be matched with two individuals who have been elected to serve as senator. In turn, each senator can be matched with one specific state that he or she represents. Both of these are real-life examples of relations.

The first value of a relation is an input value and the second value is the output value. A function is a specific type of relation in which each input value has one and only one output value. An input is the independent value, and the output value is the dependent value, as it depends on the value of the input.

Notice in the first table below, where the input is “name” and the output is “age,” each input matches with exactly one output. This is an example of a function.

Compare this with the next table, where the input is “age” and the output is “name.” Some of the inputs result in more than one output. This is an example of a correspondence that is not a function.

Let’s look back at our examples to determine whether the relations are functions or not and under what circumstances. Remember that a relation is a function if there is only one output for each input.

Relations can be written as ordered pairs of numbers or as numbers in a table of values. By examining the inputs (x-coordinates) and outputs (y-coordinates), you can determine whether or not the relation is a function. Remember, in a function each input has only one output.

There is a name for the set of input values and another name for the set of output values for a function. The set of input values is called the domain of the function. And the set of output values is called the range of the function.

If you have a set of ordered pairs, you can find the domain by listing all of the input values, which are the x-coordinates. And to find the range, list all of the output values, which are the y-coordinates.

So for the following set of ordered pairs,

[latex]\{(−2,0),(0,6),(2,12),(4,18)\}[/latex]

You have the following:

[latex]\begin{array}{l}\text{Domain}:\{−2,0,2,4\}\\\text{Range}:\{0,6,12,18\}\end{array}\\[/latex]

You try it.

In the following video we provide another example of identifying whether a table of values represents a function, as well as determining the domain and range of the sets.

In the following video we show how to determine whether a relation is a function, and define the domain and range.

Some people think of functions as “mathematical machines.” Imagine you have a machine that changes a number according to a specific rule, such as “multiply by 3 and add 2” or “divide by 5, add 25, and multiply by [latex]−1[/latex].” If you put a number into the machine, a new number will pop out the other end, having been changed according to the rule. The number that goes in is called the input, and the number that is produced is called the output.

You can also call the machine “[latex]f[/latex]” for function. If you put [latex]x[/latex] into the box, [latex]f(x)[/latex], comes out. Mathematically speaking, [latex]x[/latex] is the input, or the “independent variable,” and [latex]f(x)[/latex] is the output, or the “dependent variable,” since it depends on the value of [latex]x[/latex].

[latex]f(x)=4x+1[/latex] is written in function notation and is read “[latex]f[/latex] of [latex]x[/latex] equals [latex]4x[/latex] plus 1.” It represents the following situation: A function named [latex]f[/latex] acts upon an input, [latex]x[/latex], and produces [latex]f(x)[/latex] which is equal to [latex]4x+1[/latex]. This is the same as the equation as [latex]y=4x+1[/latex].

Function notation gives you more flexibility because you don’t have to use [latex]y[/latex] for every equation. Instead, you could use [latex]f(x)[/latex]  or [latex]g(x)[/latex] or [latex]c(x)[/latex]. This can be a helpful way to distinguish equations of functions when you are dealing with more than one at a time.

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.

Now you try it.

Analysis of the solution

We can use any letter to name the function; the notation [latex]h\left(a\right)[/latex] shows us that [latex]h[/latex] depends on [latex]a[/latex]. The value [latex]a[/latex] must be put into the function [latex]h[/latex] to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.

Let’s try another.

This would make it easy to graph both functions on the same graph without confusion about the variables.

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

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.

In the following videos we show two more examples of how to express a relationship using function notation.

Graphs of functions

When both the independent quantity (input) and the dependent quantity (output) are real numbers, a function can be represented by a graph in the coordinate plane. The independent value is plotted on the x-axis and the dependent value is plotted on the y-axis. The fact that each input value has exactly one output value means graphs of functions have certain characteristics. For each input on the graph, there will be exactly one output. For a function defined as [latex]y = f(x)[/latex], or [latex]y[/latex] is a function of [latex]x[/latex], we would write ordered pairs [latex](x, f(x))[/latex] using function notation instead of [latex](x,y)[/latex] as you may have seen previously.

We can identify whether the graph of a relation represents a function because for each x-coordinate there will be exactly one y-coordinate.

When a vertical line is placed across the plot of this relation, it does not intersect the graph more than once for any values of [latex]x[/latex].

If, on the other hand, a graph shows two or more intersections with a vertical line, then an input (x-coordinate) can have more than one output (y-coordinate), and [latex]y[/latex] is not a function of [latex]x[/latex]. Examining the graph of a relation to determine if a vertical line would intersect with more than one point is a quick way to determine if the relation shown by the graph is a function. This method is often called the “vertical line test.”

You try it.

The vertical line method can also be applied to a set of ordered pairs plotted on a coordinate plane to determine if the relation is a function.

In another set of ordered pairs, [latex]\{(3,−1),(5,−2),(3,−3),(−3,5)\}[/latex], one of the inputs, 3, can produce two different outputs, [latex]−1[/latex] and [latex]−3[/latex]. You know what that means—this set of ordered pairs is not a function. A plot confirms this.

Notice that a vertical line passes through two plotted points. One x-coordinate has multiple y-coordinates. This relation is not a function.

In the following video we show another example of determining whether a graph represents a function using the vertical line test.

Evaluate Functions

Throughout this course, you have been working with algebraic equations. Many of these equations are functions. For example, [latex]y=4x+1[/latex] is an equation that can also represent a function. When you input values for [latex]x[/latex], you can determine a single output for [latex]y[/latex]. In this case, if you substitute [latex]x=10[/latex] into the equation you will find that y must be 41; there is no other value of y that would make the equation true.

Rather than using the variable [latex]y[/latex], the equations of functions can be written using function notation. Function notation is very useful when you are working with more than one function at a time, and substituting more than one variable in for [latex]x[/latex].

Equations written using function notation can also be evaluated. With function notation, you might see a problem like this.

Given [latex]f(x)=4x+1[/latex], find [latex]f(2)[/latex]

You read this problem like this: “given [latex]f[/latex] of [latex]x[/latex] equals [latex]4x[/latex] plus one, find [latex]f[/latex] of 2.” While the notation and wording is different, the process of evaluating a function is the same as evaluating an equation: in both cases, you substitute 2 for x, multiply it by 4 and add 1, simplifying to get 9. In both a function and an equation, an input of 2 results in an output of 9.

[latex]f(x)=4x+1\\f(2)=4(2)+1=8+1=9[/latex]

You can simply apply what you already know about evaluating expressions to evaluate a function. It’s important to note that the parentheses that are part of function notation do not mean multiply. The notation [latex]f(x)[/latex] does not mean [latex]f[/latex] multiplied by [latex]x[/latex]. Instead the notation means “[latex]f[/latex] of [latex]x[/latex]” or “the function of [latex]x[/latex]” To evaluate the function, take the value given for [latex]x[/latex], and substitute that value in for [latex]x[/latex] in the expression. Let’s look at a couple of examples.

Functions can be evaluated for negative values of [latex]x[/latex], too. Keep in mind the rules for integer operations.

You may also be asked to evaluate a function for more than one value as shown in the example that follows.

Variable Inputs

So far, you have evaluated functions for inputs that have been constants. Functions can also be evaluated for inputs that are variables or expressions. The process is the same, but the simplified answer will contain a variable. The following examples show how to evaluate a function for a variable input.

In the following example, you evaluate a function for an expression. So here you will substitute the entire expression in for [latex]x[/latex] and simplify.

In the following video we show more examples of evaluating functions for both integer and variable inputs.