What are functions? What do they mean when they ask you to *evaluate* a function at a given number? In this article, we’ll see what functions are all about and how to work with them. These skills will help you to succeed on the GED Mathematics section.

## Functions — Input and Output

Let’s start with a simple example to help explain the terms and concepts. Consider the following **function**:

*f*(*x*) = 3*x* + 1

You can tell that we’re dealing with a function because of the notation “*f*(*x*).” This notation tells you that the function name is *f* and the input variable is *x*. The expression that follows, 3*x* + 1, is the function definition or rule. That tells you what the function does to its input.

When you **evaluate** a function at a given number, you simply replace the input variable with that number and work out the numerical value. (This is often called **plugging in**.)

The notation tells you what to plug in. If you see *f*(2), then that means you have to substitute 2 in place of *x* in the function and work out the output value.

For example, let’s find *f*(2) if *f*(*x*) = 3*x* + 1.

*f*(2) = 3(2) + 1 = 6 + 1 = 7.

Pretty easy, right?

## Tables and Graphs of Functions

We often use the variable *y* to stand for the output of a function. So we might write *y* = *f*(*x*).

Suppose you want to evaluate a function at multiple input values. You might set up a table to keep track of all of the input and output.

For example, let’s evaluate *f*(*x*) = 3*x* + 1 at -2, -1, 0, 1, and 2.

x | y |
---|---|

-2 | 3(-2) + 1 = -5 |

-1 | 3(-1) + 1 = -2 |

0 | 3(0) + 1 = 1 |

1 | 3(1) + 1 = 4 |

2 | 3(2) + 1 = 7 |

Now that we have a table a values, we can go one step further and draw a **graph** of the function. A graph is a visual display of the input and output values. When you plot the points of a graph, always be sure that your coordinates are in the order (*x*, *y*), or in other words, (*x*, *f*(*x*)).

### Using the Graph to Evaluate a Function

In fact, the graph can be used to evaluate a function even if you don’t have the function definition explicitly given.

For example, the graph below represents a function *y* = *g*(*x*). What is the value of *g*(5)?

This time, we can’t plug *x* = 5 into anything, because there’s no function definition provided. To solve this, just look for the *y* coordinate when *x* = 5. Based on the graph, we find that *g*(5) = 4.

Graphing is a key topic that shows up in a number of forms on the GED Math section. Check out the following article for a discussion of graphing from a different perspective: GED Math: Data Analysis Questions.

## Evaluating Functions at Variable Quantities

Sometimes a problem will ask you to evaluate an expression like *f*(*a* + 1) or *f*(*x*^{2}). These are more difficult because it’s no longer just “plug in and determine a value.” Instead, you’ll have to use your Algebra skills to simplify the expression. Let’s see how this works by example.

Evaluate *f*(*d* – 4) if *f*(*x*) = 3*x*^{2} – *x* + 9.

The first step is similar to what we did above. Replace *x* by the input quantity. However, instead of a number, you have to “plug in” a variable expression, (*d* – 4) in this case. Be sure to put parentheses around the expression everywhere in the function.

*f*(*d* – 4) = 3(*d* – 4)^{2} – (*d* – 4) + 9

Next, carefully simplify the result. Start by squaring the binomial in the first term.

= 3(*d*^{2} – 8*d* + 16) – (*d* – 4) + 9

Then use the Distributive Property. Be careful with that minus!

= 3*d*^{2} – 24*d* + 48 – *d* + 4 + 9

Finally combine like terms.

= 3*d*^{2} – 25*d* + 61

Although it might seem complicated at first, this is just an extension of what you already know from Algebra.

## Setting up Functions

Sometimes in a word problem, you may have to set up a function and then evaluate it. For example, suppose that your smartphone provider charges you $25 per month plus $15 per each gigabyte of data used. Write the function that represents the total monthly bill when *x* gigabytes of data are used.

First think it out! If you didn’t use any data in a given month then you’d still have to pay the monthly fee of $25. Using one gigabyte would raise that bill to $25 + $15 = $40. Two gigabytes would cost $25 + $15 · 2 = $55, and three would cost $25 + $15 · 3 = $70, etc.

So the pattern is: $25 + $15 · (gigabytes used). Or, in function notation, with *x* = *gigabytes of data*, you would write your answer as follows:

*f*(*x*) = 25 + 15x

That’s all there is to it!

## Practice Problems

Ok, are you ready to work a few problems yourself? The following problems are modeled from actual GED Math questions. Good luck!

- Find the value of
*h*(-2) if*h*(*x*) = 3*x*^{2}– 2*x*+ 5. - A car sales representative gets paid $800 per week plus 10% of any sales she makes during that week. Set up a function that calculates the sales representative’s weekly pay, using
*s*for the total sales (in dollars). - Evaluate
*g*(*t*^{3}) if*g*(*t*) =*t*^{4}– 5*t* - Which function best represents the ordered pairs of data shown in the table?
A.

*f*(*x*) = 2*x*– 1 B.*f*(*x*) = 5*x*– 7 C.*f*(*x*) =*x*^{2}– 1 D.*f*(*x*) = ½(*x*+ 1)^{2}

A. -11 B. -3 C. 13 D. 21

A. 800 + 0.1*s* B. 0.1 + 800*s* C. 800*s*^{0.1} D. 800.1*s*

A. *t*^{3} – 5*t* B. *t*^{12} – 5*t*^{3} C. *t*^{7} – 5*t*^{3} D. (*t*^{4} – 5*t*)(*t*^{3})

#### Answer Key

1. D 2. A 3. B 4. C