Function notation

A new way to write y

Function notation writes the output of a function as f(x), read "f of x". Instead of y = 3x + 2, we write:

f(x) = 3x + 2

• f is the function name.
• x is the input.
• f(x) is the output.

It does not mean f times x.

Why use it?

Function notation tells you both the input and the output in one expression:

• "y = 17 when x = 5" becomes simply f(5) = 17.
• You can name several functions: f(x), g(x), h(t), C(n), so it’s clear which is which.

Evaluating functions

To evaluate, replace x with the given value.

f(x) = 3x + 2

Expressions as inputs

The argument (inside the brackets) can be an expression.

g(x) = x² − 1

• g(a) = a² − 1
• g(2a) = (2a)² − 1 = 4a² − 1
• g(x + 1) = (x + 1)² − 1 = x² + 2x

Working backwards

Example: f(x) = 3x + 2. For what x is f(x) = 20?

• 3x + 2 = 20 → x = 6

Reading from a graph

f(3) means "go to x = 3 and read the y-value". If the graph passes through (3, 7), then f(3) = 7.

Reading from a table

h(2) = −3. And h(x) = 5 when x = −2.

Real-life notation

• C(n) = 15n + 40: cost for n people. C(10) = $190.
• h(t) = −5t² + 20t: height of a ball after t seconds. h(2) = −20 + 40 = 20 m.

Worked examples

Example 1: f(x) = 2x² − x. Find f(3). → 18 − 3 = 15.

Example 2: g(x) = 7 − 4x. Find g(−2). → 7 + 8 = 15.

Example 3: p(x) = 5x − 1. Solve p(x) = 24. → x = 5.

Common mistakes

• Reading f(x) as f × x.
• Not using brackets when substituting negatives: f(−3) for x² is (−3)² = 9.
• Confusing f(2) = 7 with f(7) = 2.

Try it yourself

1. f(x) = 4x − 3. Find f(5).
2. g(x) = x² + 2x. Find g(−3).
3. h(x) = 10 − x. Find h(a + 2).
4. f(x) = 2x + 9. Solve f(x) = 1.
5. A graph passes through (−1, 6). Write this in function notation.

Answers: 1) 17 2) 3 3) 8 − a 4) −4 5) f(−1) = 6

Quick recap

• Function notation f(x) shows the output for input x.
• Evaluate by substituting into the rule.
• f(a) = b means the point (a, b) is on the graph.