Linear after quadratic
For f(x) = 2x + 1 and g(x) = x² − 3, find f(g(x)).
- Insert x² − 3 into the input of f.
- Expand 2(x² − 3) + 1.
Answer: 2x² − 5
Composite functions
Work out f(g(x)) and g(f(x)), keeping brackets around the function you substitute.
Before you start
You’ll need Function notation and evaluation, Expanding and factorising. Follow a link if you want to revise it first.
Which function comes first?
f(g(x)) means apply g first and f second. Read from the inner brackets outward.
Replace every input of f by the entire expression g(x); brackets protect its structure.
The output of the inner function must belong to the domain of the outer function.
The examples below use the same functions in both orders. You’ll get a different polynomial each time.
Calculating the two compositions
Quadratic after linear
With the same functions, find g(f(x)).
- Insert 2x + 1 into the input of g.
- Expand (2x + 1)² − 3, including the middle term.
Answer: 4x² + 4x − 2
Applying f first when asked for f(g(x)) reverses the order.
Practice questions
The first link uses smaller values. Move on when you can answer without referring to the examples. The difficulty settings aren’t tied to exam grades.
Practice is filtered to your selected course. Change course or options when needed.
Check your understanding
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Evaluate both composite functions at x = 0. Does that settle whether they are identical?