Functions questions on the PSAT/NMSQT Math section test whether you understand what a function actually represents — a rule that pairs each input with exactly one output — and whether you can work fluently with that rule in both directions: evaluating a function at a given input, and working backward from a known output to find the input that produced it. The exam also tests function notation tricks, like substituting an entire expression (not just a number) as the input, and recognizing when "f(x+1)" and "f(x)" describe the same rule shifted by one step.
The 7 problems below cover domain identification, reading a mapping diagram, quadrant reasoning with coordinate points, and several function-notation substitution problems where the input itself is an algebraic expression. Every problem includes a complete step-by-step explanation, hidden by default so you can test yourself honestly before checking your work. For problems where you're given f(x+1) or a similar shifted expression and asked for a different input value, the key move is setting the inside expression equal to the target input first, solving for the helper variable, and only then substituting back into the given formula. For unlimited additional practice, our free PSAT/NMSQT Math QBank has over 2,500 more problems across every PSAT/NMSQT Math topic, each with instant explanations.
What is the domain of the function that contains points at (−5,2), (−2,1), (0,2), and (4,−3)?
The domain of a function is the set of all input values, which are the x-coordinates of each point.
The x-coordinates of the given points are −5, −2, 0, and 4. The correct answer is D.
Which of the following relations is a correct representation of the mapping shown above?
Trace each arrow from the x-column to the y-column, keeping x listed first in each ordered pair (matching the direction the mapping diagram shows).
Options C and D list y before x, which reverses the mapping's direction — a common trap for this question type. Tracing the arrows correctly gives −5→8, −2→7, 2→−1, and 5→4. The correct answer is B.
If point (7,b) is in Quadrant I and point (a,−3) is in Quadrant III, in which Quadrant is the point (a,b)?
Quadrant I requires both coordinates positive. Since (7,b) is in Quadrant I and 7 is already positive, b must be positive too.
Quadrant III requires both coordinates negative. Since (a,−3) is in Quadrant III and −3 is already negative, a must be negative too. That makes (a,b) = (negative, positive), which is Quadrant II. The correct answer is B.
If f(x) = −2x + 7, what is f((1/2)x + 3) equal to?
Substitute the entire expression (1/2)x + 3 in place of x in the function: f((1/2)x+3) = −2((1/2)x+3) + 7.
Distribute: −2(1/2)x − 2(3) + 7 = −x − 6 + 7 = −x + 1. The correct answer is A.
g(x) = kx³ + 3. For the function g defined above, k is a constant and g(−1) = 5. What is the value of g(1)?
Use g(−1) = 5 to find k: k(−1)³ + 3 = −k + 3 = 5, so k = −2.
Now evaluate g(1): g(1) = (−2)(1)³ + 3 = −2 + 3 = 1. The correct answer is C.
If f(x+1) = −(1/2)x + 6, what is the value of f(−3)?
Find the value of x that makes the input equal −3: set x + 1 = −3, so x = −4.
Substitute x = −4 into the given formula: f(−3) = −(1/2)(−4) + 6 = 2 + 6 = 8.
f(x) = x² − b. In the function above, b is a constant. If f(−2) = 7, what is the value of f(b)?
Use f(−2) = 7 to find b: (−2)² − b = 4 − b = 7, so b = −3.
Now evaluate f(b) = f(−3): (−3)² − b = 9 − (−3) = 9 + 3 = 12.
These 7 problems are just a sample. Practice thousands more PSAT/NMSQT Math problems, organized by topic, with instant explanations — completely free.
Try the Free PSAT/NMSQT Math QBank →