PSAT/NMSQT Math: Functions
A function is a rule that assigns exactly one output to each input, and the PSAT/NMSQT tests this idea from every angle: evaluating functions directly, reading values off a graph or table, interpreting what function notation means in a real-world context, and working with inverses and compositions. This free, complete lesson covers all of these skills. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quizzes below.
1. Evaluating Functions
To evaluate a function f(x) at a specific value, substitute that value everywhere x appears in the function's rule, then simplify. The input doesn't have to be a plain number, it can also be a variable expression, which you substitute in exactly the same way.
Worked Example: If f(x) = 3x − 4, find f(5).
f(5) = 3(5) − 4 = 15 − 4 = 11
1. If f(x) = 2x² − 3, find f(−3).
Explanation: f(−3) = 2(−3)² − 3 = 2(9) − 3 = 18 − 3 = 15.
2. If f(x) = 4x + 1, find f(3a − 2).
Explanation: Substitute (3a − 2) for x: 4(3a − 2) + 1 = 12a − 8 + 1 = 12a − 7.
3. If f(x) = 5x − 3 and f(x) = 22, what is the value of x?
Explanation: 5x − 3 = 22 → 5x = 25 → x = 5.
4. Given f(x) = 3x + k and f(5) = 29, what is the value of k?
Explanation: f(5) = 3(5) + k = 15 + k = 29, so k = 14.
2. Reading Functions from Graphs, Tables, and Context
A function can be given as a graph, a table of values, or a description of a real-world situation, not just an algebraic rule. In all these cases, f(a) = b means the point (a, b) is part of the function, whether that point appears on a graph, in a table row, or as a described result.
Worked Example: A cost function satisfies C(10) = 35. What does this tell you in context, if C(x) models the cost, in dollars, of producing x units?
Producing 10 units costs $35
1. A graph of g passes through the points (0, 3), (2, 7), and (4, 15). What is g(2)?
Explanation: The point (2, 7) directly tells us that when x = 2, the output is 7, so g(2) = 7.
2. The following table shows values of f(x). What is f(x) + 3 when x = 1?
Explanation: If the table shows f(1) = 6, then f(1) + 3 = 6 + 3 = 9. Always read the specific value directly from the table row before doing any further arithmetic.
3. A ball's height over time is modeled by a function h(t), where t is measured in seconds after being thrown. What does the y-intercept, h(0), represent?
Explanation: Since t = 0 represents the very start, h(0) gives the height at that starting instant, before any time has passed.
4. A depreciation graph shows a laptop's value over time. At t = 3 years, the graph shows a value of about $450. What does this represent?
Explanation: Reading a point off a graph gives the output (value) at that specific input (time), so this point means the laptop's predicted value is about $450 at the 3-year mark.
3. Inverse Functions
The inverse of a function, written f−1(x), reverses the roles of input and output: if f(a) = b, then f−1(b) = a. To find an inverse algebraically, swap x and y in the equation and solve for the new y.
Worked Example: Find the inverse of f(x) = x − 5.
Write y = x − 5, swap x and y: x = y − 5, solve for y:
f−1(x) = x + 5
1. Find the inverse of f(x) = 2x + 6.
Explanation: y = 2x + 6 → swap: x = 2y + 6 → x − 6 = 2y → y = (x − 6)/2.
2. If f(3) = 11, what must be true about f−1?
Explanation: An inverse function reverses input and output: since f sends 3 to 11, f−1 must send 11 back to 3.
3. Find the inverse of f(x) = x4 − 1.
Explanation: y = x/4 − 1 → swap: x = y/4 − 1 → x + 1 = y/4 → y = 4(x + 1).
4. Function Composition
Composing two functions, written f(g(x)), means evaluating the inner function g first, then using that entire result as the input to the outer function f. Work from the inside out, one function at a time.
Worked Example: If f(x) = x + 3 and g(x) = 2x, find f(g(4)).
Evaluate the inner function first: g(4) = 2(4) = 8.
f(g(4)) = f(8) = 8 + 3 = 11
1. If f(x) = x² and g(x) = x + 1, find f(g(3)).
Explanation: g(3) = 3 + 1 = 4. Then f(4) = 4² = 16.
2. If f(x) = 2x + 1 and g(x) = x² − 1, find f(g(2)) − g(f(2)).
Explanation: g(2) = 3, f(g(2)) = f(3) = 7. f(2) = 5, g(f(2)) = g(5) = 24. So f(g(2)) − g(f(2)) = 7 − 24 = −17.
3. For what x-values is the function g(x) = (x² − 25)/(x + 9) undefined?
Explanation: A rational function is undefined wherever its denominator equals 0: x + 9 = 0 → x = −9.
Common Mistakes to Avoid
- Confusing f(a) = b with f(b) = a. The input always goes inside the parentheses; the output is the result after evaluating.
- Evaluating a composition in the wrong order. In f(g(x)), always evaluate g first, then plug that entire result into f, never the reverse.
- Forgetting to swap x and y when finding an inverse. Simply solving the original equation for y again does not produce the inverse function.
- Misreading which axis or column gives the input versus the output on a graph or table. The input (x) determines which row or point to look at; the output (y) is the value found there.
- Ignoring domain restrictions on rational or radical functions. A function is undefined wherever its denominator is 0, or (for even roots) wherever the expression inside would be negative.
Frequently Asked Questions
What's the difference between f(x) and f−1(x)?
f(x) is the original function; f−1(x) is its inverse, which undoes what f does. If f turns 3 into 11, then f−1 turns 11 back into 3.
How do I evaluate a function given only a graph, with no equation?
Find the input value on the x-axis, trace straight up or down to the curve, and read the corresponding y-value where the curve crosses that vertical line. That y-value is the function's output.
Where can I practice more problems like these?
The Functions quizzes in the PSAT/NMSQT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.