Function Notation

Grade 6

What f(x) means

Function notation is a compact way to describe a rule. f(x) = 2x + 1 means: take any number x, double it, add 1. Writing f(3) means apply the rule to x = 3: f(3) = 7.

You may see g(x), h(x) or other letters — the letter is just the function's name. The working inside is what matters.

Composite functions — functions inside functions

The composite function fg(x) means "do g first, then apply f to the result". It is read right-to-left: fg(x) = f(g(x)).

EXAMPLE

f(x) = 3x + 1 and g(x) = x². Find fg(2) and gf(x).

fg(2): first g(2) = 4, then f(4) = 13. fg(2) = 13. gf(x): first f(x) = 3x + 1, then g(3x + 1) = (3x + 1)². gf(x) = (3x + 1)².

Inverse functions

The inverse function f⁻¹(x) reverses what f does. To find it: write y = f(x), swap x and y, then rearrange to make y the subject. That gives you f⁻¹(x).

EXAMPLE

Find the inverse of f(x) = (x − 3) / 2.

Write y = (x − 3)/2. Swap x and y: x = (y − 3)/2. Multiply by 2: 2x = y − 3. Add 3: y = 2x + 3. So f⁻¹(x) = 2x + 3.

Order matters in composites

fg(x) and gf(x) are generally different. Always apply the rightmost function first. Writing it out as f(g(x)) makes the order explicit and avoids confusion.

Try this one

The diagram shows the graph of \(y = q(x)\) with a maximum at \((-1, 5)\) and passing through \((0, 3)\).

(a) Write down the coordinates of the maximum of \(y = q(x) + 2\). [1]

(b) Write down the coordinates of the maximum of \(y = q(2x)\). [2]

Nova's hint:

This involves vertical and horizontal transformations of a graph.

Ask yourself: 'Does this change affect x or y values, and does it stretch, shift, or both?'

For y = q(2x), the x-coordinates are halved (compressed horizontally) while y-values stay the same.

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