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Functions for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT tests about functions in one place: function notation, evaluating functions, domain and range, the vertical line test, composite functions, and transformations — with step-by-step examples, worked problems, and 2 free practice quizzes.
Practice these quizzes
Free · No signupWhat is a function?
A function is a rule that assigns exactly one output to every input. Function notation f(x) simply means "the output of function f when the input is x" — it's another name for y.
The defining feature of a function is that each input produces exactly one output. If a single x-value ever maps to two different y-values, the relationship is not a function.
Evaluating functions
If f(x) = 3x² − 4, what is f(2)?
If the PSAT asks for the value of x when f(x) equals a given number, set the whole expression equal to that number and solve for x — the reverse of a normal evaluation.
Domain & range
| Term | Definition |
|---|---|
| Domain | The set of all possible input (x) values |
| Range | The set of all possible output (y) values |
What values of x are excluded from the domain of f(x) = 5 / (x − 3)?
Two common domain restrictions: a denominator can never equal zero, and the expression under a square root can never be negative. Check both before assuming a domain is "all real numbers."
The vertical line test
The vertical line test checks whether a graph represents a function: if any vertical line crosses the graph more than once, the graph is not a function, since that x-value would have more than one output.
A circle fails the vertical line test (most x-values hit the circle twice), which is why a circle's equation is not itself a function — even though it's a perfectly valid graph.
Composite functions
A composite function, written f(g(x)), means: first evaluate g(x), then plug that result into f.
If f(x) = 2x + 1 and g(x) = x², find f(g(3)).
f(g(x)) is evaluated from the inside out — always work out the innermost function first. Reversing the order (evaluating f before g) gives a completely different, incorrect result.
Transformations of functions
| Change to f(x) | Effect on the graph |
|---|---|
| f(x) + k | Shifts up (k > 0) or down (k < 0) |
| f(x − h) | Shifts right (h > 0) or left (h < 0) |
| −f(x) | Reflects over the x-axis |
Horizontal shifts move opposite the sign inside the parentheses: f(x − 3) shifts the graph right by 3, even though the sign is negative.
Common PSAT traps
- Confusing f(x) with x · f: f(x) is not multiplication — it's function notation meaning "f evaluated at x."
- Evaluating a composite function in the wrong order: always work from the innermost function outward.
- Missing a domain restriction: check for zero denominators and negative values under square roots.
- Getting horizontal shift direction backward: the shift direction is opposite the sign shown inside the parentheses.
Test day strategy for functions
| Question signal | Fastest approach |
|---|---|
| f(a) = ? question | Substitute a directly into the function |
| Solve for x when f(x) = a given value | Set the function expression equal to that value, then solve |
| Domain restriction question | Check for zero denominators and negative square root expressions |
| f(g(x)) composite question | Evaluate the inner function first, then substitute into the outer function |
| Graph shifted on the coordinate plane | Match the shift to the transformation rules, checking signs carefully |
Now put it to work
Two quiz sets, each building on the last — start with Quiz 1 and work through in order, or jump straight to the topic you need.
Functions 1
Function notation, evaluating functions, domain and range.
Start quiz → Quiz 2Functions 2
Composite functions, transformations, and interpreting graphs.
Start quiz →