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|
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.
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.
|STRATEGIC THINKING||MATH SCRATCHWORK|
|Step 1: Read the question, identifying and organizing important information as you go
The question is asking for the answer choice that could not be in the range of this function.
Step 2: Choose the best strategy to answer the question
How can you determine which of the four choices is not part of the range?
You’ll need to examine each piece of the domain individually to learn more.
First, summarize each piece of the domain description.
Can you use the domain information to eliminate any answer choices?
What happens when x > −1?
Because the range is 0 when x > −1, you can eliminate B.
Would your knowledge of number properties be helpful for the second domain component?
Absolutely. Because the range x² − x only applies when x ≤ −1, you only need to consider what happens when a negative number less than −1 is substituted for x.
No matter what number you use, your output will always be positive. Eliminate C and D.
|domain x ≤ −1: range = x² − x domain x > −1: range = 0 (negative)² − negative → positive + positive → positive|
|Step 3: Check that you answered the right question
The range of f(x) consists of only positive numbers, so (A) is correct.
|−4 is not positive|
You might be tempted to plug the answer choices in and solve for x, but this will cost you valuable time. While Backsolving can be a strategy of last resort on problems like this, it takes far too long. Use it only if you can’t approach the problem in any other way. The PSAT will reward you for knowing the quickest way to answer the question, which in this case involves knowing number properties.