# PSAT Math: Functions

Functions act as rules that transform inputs into outputs, and they differ from equations in that each input must have only one corresponding output. For example, imagine a robot: Every time you give it an apple, it promptly cuts that apple into three slices. The following table summarizes the first few inputs and their corresponding outputs.

 Domain, x: # apples given to robot Range, f (x): # slices returned by robot 0 0 1 3 2 6 3 9

From the table you see that the output will always be triple the input, and you can express that relationship as the function f (x) = 3x (read “f of x equals three x”).

PSAT questions, especially those involving real-world situations, might ask you to derive the equation of a function, so you’ll need to be familiar with the standard forms. Following is the standard form of a linear function:

f (x) = kx + f (0)

The input, or domain, is the value represented by x. Sometimes the domain will be constrained by the question (e.g., x must be an integer). Other times, the domain could be defined by real-world conditions. For example, if x represents the time elapsed since the start of a race, the domain would need to exclude negative numbers. The output, or range, is what results from substituting a domain value into the function and is represented by f(x). The initial amount, or y-intercept, is represented by f(0)—the value of the function at the very beginning. If you think this looks familiar, you’re absolutely right. It’s just a dressed-up version of the standard y = mx + b equation you’ve already seen. Take a look at the following table for a translation:

 Function Notation What It Represents Slope-Intercept Counterpart f(x) dependent variable or output y k rate of change, slope m f(0) y-intercept or initial quantity in a word problem b

As you might have guessed, an exponential equation has a standard function notation as well. Here we’ve used g in place of f for visual clarity. Know that the letter used to represent a function (f, g, h, etc.) is sometimes arbitrarily chosen.

g(x) = g(0)(1+ r)x

Just as before, g(0) represents the initial amount and r represents the growth (or decay) rate. Recognizing that function notation is a variation of something you already know will go a long way toward reducing nerves on Test Day. You should also note that graphing functions is a straightforward process: In the examples above, just replace f(x) or g(x) with y and enter into your graphing calculator.

Note

A quick way to determine whether an equation is a function is to conduct the vertical line test: If a vertical line passes through the graph of the equation more than once for any given value of x, the equation is not a function.

Below is an example of a test-like question about functions.

Use the Kaplan Method for Math to solve this question, working through it step-by-step. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.