PSAT Math: Advanced Functions

The ability to interpret the graph of a function will serve you well on Test Day. To interpret graphs of functions, you’ll need to utilize the same skills you use to interpret “regular” equations on the coordinate plane, so this material shouldn’t be completely foreign.

You should know that a function is merely a dressed-up equation, so translating from function to “regular” notation or vice versa is a straightforward process. Consider the following brief example.

Suppose you’re asked to find the value of x for which f(x) = 6. Because f(x) represents the output value, or range, translate this as “When does the y value equal 6?” To answer the question, find 6 on the y-axis, then trace over to the function (the line). Read the corresponding x value: It’s −2, so when f(x) = 6, x must be −2.

The PSAT might also present functions in the form of tables. These may or may not have an equation associated with them, but regardless, you’ll need to be adept at extracting the information necessary to answer questions. Most of the time the table will have just two columns, one for the domain and another for the range.

Expert Tip

Remember: A value of f(x) corresponds to a location along the y-axis. A value of x corresponds to a location on the x-axis.

Now let’s try a test-like example.

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

There are several ways in which the PSAT might ask you to juggle multiple functions simultaneously. Fortunately, the rules governing what to do are easy to understand. To start, we’ll look at how to combine functions. This technique simply involves adding, subtracting, multiplying, and/or dividing the functions in play. Check out the following table for a synopsis of how to combine functions with the four basic operations (and make them look less intimidating).

On Test Day, you might see piecewise functions. A piecewise function is a function that is defined, literally, by multiple pieces. What breaks a function into pieces are different rules that govern different parts of the function’s domain. Here’s an example:

In the function shown, the behavior of the graph depends on the domain. Linear, quadratic, and even a single point interval make up this function. Each “rule” is written inside the open bracket in “pieces.” To the right is the domain interval for which each “rule” applies. On the graph, an open dot indicates that a point is not included in the interval; a closed dot indicates one that is. Note that the different inequality signs used in the domain constraints dictate whether a dot is open or closed on the graph. For a single-point interval, an equal sign is used.

To evaluate a piecewise function, first determine to which piece of the domain the input value belongs, and then substitute the value into the corresponding rule. For example, in the function above, f(2) = (2)² = 4, because the input value 2 is between –2 and 3 (the second piece of the domain).

Similarly, f(5) = –3(5) + 11 = – 4 because the input value 5 is greater than 3 (the last piece of the domain). You can confirm these values by looking at the graph. At x = 2, the point on the graph is (2, 4), and at x = 5, the point on the graph is (5, – 4).

Give this function question a try:

Appearances can be deceiving. At first glance, this question looks tough, but the following table will clarify anything confusing. Kaplan’s strategic thinking is on the left, along with suggested math scratchwork on the right.