PSAT/NMSQT Prep 2022 - Eggert M.D., Strelka A. 2022

Function notation
Functions
Passport to advanced math

LEARNING OBJECTIVES

After this lesson, you will be able to:

· Apply function notation

· Define the domain and range of a function

· Evaluate the output of a function for a given input

To answer a question like this:

The function h(x) is defined above. Out of the statements below, which must be true about h(x)?

I. h(8) = 2.5

II. The domain of h(x) is all real numbers.

III. h(x) may be positive or negative.

A. I and II

B. I and III

C. II and III

D. I, II, and III

You need to know this:

A function is a rule that generates one unique output for a given input. In function notation, the x-value is the input and the y-value, designated by f(x), is the output. (Note that other letters besides x and f may be used.)

For example, a linear function has the same form as the slope-intercept form of a line; f(x) is equivalent to y:

In questions that describe real-life situations, the y-intercept will often be the starting point for the function. You can think of it as f (0), or that value of the function where x = 0.

The set of all possible x-values is called the domain of the function, while the set of all possible y-values is called the range.

You need to do this:

· To find f(x) for some value of x, substitute the concrete value in for the variable and do the arithmetic.

· For questions that ask about the domain of a function, check whether any inputs are not allowed, for example, because they cause division by zero.

· For questions that ask about a function of a function, such as g(f(x)), start on the inside and work your way out.

Explanation:

Check each statement. For the first statement, plug in 8 for x:

So the first statement is true. Eliminate choice (C).

For the second statement, you need to determine the set of all permitted x-values for this function. Note that the function will be undefined at x = 2 (because at x = 2, the denominator would be zero). Thus, 2 is not a permitted x-value, and the domain is not all real numbers. The second statement is false. Eliminate (A) and (D).

By process of elimination, the answer is (B), and on test day, you would stop here. For the record, here’s why the third statement is true: you’ve already established that h(8) = 2.5, so h(x) can be positive. Try a smaller value for x, such as zero, to get a negative value for h(x): , so h(x) can be negative as well, which means statement III is true.

Try on Your Own

Directions: Take as much time as you need on these questions. Work carefully and methodically. There will be an opportunity for timed practice at the end of the chapter.

HINT: Are there any answer choices in Q1

you can eliminate right away?

1. If f(x) = x² − x for all x ≤ −1 and f(x) = 0 for all x > −1, which of the following could NOT be a value of f(x)?

A. − 4

B.0