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 two curves. Curve A has equation y = f(x) and curve B is a transformation of curve A.

Curve B passes through (2, 1) and curve A passes through (2, 4) at the same x-value.

Express the equation of curve B in terms of f(x). [2]

Nova's hint:

This involves identifying a vertical stretch from the relationship between the y-values of two curves at the same x.

Ask yourself: 'If I know f(2) = 4 and the transformed curve gives 1 at x = 2, what scale factor multiplies f(x) to give that result?'

Try to Solve it with Nova?