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) = 3x + 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, 3x + 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) = 3x + 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) = 3x + 1 at -2, -1, 0, 1, and 2.
|-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(x2). 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) = 3x2 – 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(d2 – 8d + 16) – (d – 4) + 9
Then use the Distributive Property. Be careful with that minus!
= 3d2 – 24d + 48 – d + 4 + 9
Finally combine like terms.
= 3d2 – 25d + 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!
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) = 3x2 – 2x + 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(t3) if g(t) = t4 – 5t
- Which function best represents the ordered pairs of data shown in the table?
A. f(x) = 2x – 1 B. f(x) = 5x – 7 C. f(x) = x2 – 1 D. f(x) = ½(x + 1)2
A. -11 B. -3 C. 13 D. 21
A. 800 + 0.1s B. 0.1 + 800s C. 800s0.1 D. 800.1s
A. t3 – 5t B. t12 – 5t3 C. t7 – 5t3 D. (t4 – 5t)(t3)
1. D 2. A 3. B 4. C