Read function notation, evaluate f(x) at a given input, and connect the algebra to what the graph actually looks like. The notation trips up more students than the arithmetic does. Every question comes with a written walkthrough of exactly what each piece of f(x) means.
Evaluating f(x) = x² + 3x at x=2 means replacing every x, including inside the square — f(2) = 2² + 3(2), not 2² + 3x.
Shifting the input changes what goes into the function; adding to the output shifts the result afterward. They produce different expressions and different graphs.
Finding f(3) means starting on the x-axis at 3 and reading up to the graph; solving f(x)=3 means starting on the y-axis at 3 and reading across — mixing up the two gives the wrong axis entirely.
Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.
If f(x) = 3x² − 4x + 1, find f(2).
A — 5. Substitute 2 for every x: 3(2)² − 4(2) + 1 = 3(4) − 8 + 1 = 5. 29 (B) comes from squaring the whole term 3x as (3×2)² = 36 instead of squaring just x first: 3×(2²) = 12. 21 (C) comes from treating −4x as +4x, flipping the sign on the middle term.
If f(x) = 2x + 5, find f(4) − f(1).
A — 6. Evaluate separately: f(4) = 2(4)+5 = 13 and f(1) = 2(1)+5 = 7, so f(4) − f(1) = 6. 11 (B) comes from computing f(4−1) = f(3) = 2(3)+5 = 11 instead — treating f(a) − f(b) as if it were f(a − b), which doesn't hold for this function.
The graph of f passes through (2, 7) and satisfies f(x) = f(x−1) + 3 for all x. What is f(5)?
A — 16. Apply the recursive rule three times from the known value: f(3) = f(2)+3 = 10, f(4) = f(3)+3 = 13, f(5) = f(4)+3 = 16. 19 (B) comes from adding 3 four times instead of three — miscounting the number of steps from x=2 to x=5. 13 (C) stops one step early, at f(4) instead of f(5).
Function notation is a translation skill before it's an algebra skill — once f(x) reads naturally, the arithmetic underneath is usually the easy part.
Before plugging in a number, say out loud what you're computing — "the output when the input is 2." Untimed practice is where the notation stops feeling foreign.
Move to timed sessions once substitution is automatic, including when the input is an expression, not just a number.
Pair evaluating functions with linear and quadratic functions in a mock — the notation is identical, only the rule inside f changes.
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