Functions • Topic 1 of 4

Function Notation

The notation f(x) names a rule: f is the rule, x is the input, and f(x) is the output. Reading "f(3)" means "run the rule on 3", not "multiply f by 3" — the parentheses do not mean multiplication here. So if f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7. The input can itself be an expression or another variable: f(a + 1) means substitute the entire quantity (a + 1) for x wherever x appears. A common GRE twist gives you f(x) and an output and asks for the input — solve f(x) = 15 as an equation. Another asks you to compare f(a) and f(b) as a Quantitative Comparison, which you handle by substituting, not by guessing. The letter used is cosmetic; g(t) or h(n) work identically. Keep every substitution wrapped in parentheses and most "function" difficulty disappears.

A function machine taking input 3 through the rule f(x) = 2x + 1 to output 7A function is a rule: input → f → output3input xx = 3ff(x) = 2x + 12(3)+17output f(3)f(3) = 2(3) + 1 = 7

✅ Solved examples

1. If f(x) = 3x − 4, find f(5).
Replace x with 5: f(5) = 3(5) − 4 = 15 − 4 = 11.
2. If f(x) = x² − 2x, find f(−3).
f(−3) = (−3)² − 2(−3) = 9 + 6 = 15. (The −2 times −3 is +6 — the sign is the usual trap.)
3. If f(x) = 2x + 7 and f(a) = 21, find a.
Set the rule equal to 21: 2a + 7 = 21, so 2a = 14 and a = 7.
4. If f(x) = x² + 1, find f(a + 1).
Substitute the whole quantity: f(a + 1) = (a + 1)² + 1 = a² + 2a + 1 + 1 = a² + 2a + 2.

✏️ Practice — try these, take hints as needed

1. If f(x) = 4x − 9, find f(6).
Replace x with 6.
4(6) − 9.
24 − 9.
15
2. If g(x) = x² + 3x, find g(−4).
Substitute −4 in parentheses.
(−4)² + 3(−4).
16 − 12.
4
3. If f(x) = 5x − 2 and f(k) = 33, find k.
Set 5k − 2 = 33.
5k = 35.
Divide by 5.
k = 7
4. If f(x) = 2x² − x, find f(a − 1) in expanded form.
Substitute (a − 1) for x.
2(a − 1)² − (a − 1).
Expand 2(a² − 2a + 1) then subtract (a − 1).
2a² − 5a + 3
5. If h(t) = 10 − t², find h(3) − h(1).
Find each value separately.
h(3) = 10 − 9 = 1; h(1) = 10 − 1 = 9.
Subtract.
1 − 9 = −8

📝 Topic test — 8 questions

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