Functions: worked exam questions

Composite Functions and Solving fg(x) = k

Form fg(x) and gf(x), see why order matters, then solve fg(x) = 21, a 0606 functions question that rewards careful substitution.

Domain of a Composite Function

Form a composite involving a square root and find its domain, a 0606 functions question testing which inputs keep the composite defined.

Finding an Inverse Function and Its Domain

Find the inverse of f(x) = 3/(x − 2) and state its domain, a 0606 question testing the swap-and-rearrange method and the domain–range link.

Inverse of a Rational Function

Find the inverse of a rational function like (2x+1)/(x−3), a 0606 question testing the collect-and-factor step that isolates x.

Range and Why a Function Is Not One-One

Find the greatest value and range of 5 − (x − 2)², explain why it is not one-one, and find the domain restriction that makes it invertible, a 0606 functions question.

Range of a Composite Function on an Interval

Form a composite function and find its range over a restricted domain, a 0606 functions question combining composition with interval reasoning.

Showing a Function Is Self-Inverse

Find the inverse of a rational function and show it equals the original, a 0606 functions question on self-inverse functions where f⁻¹(x) = f(x).

Solving a Composite Equation gf(x) = k

Form a composite and solve it equal to a value, a 0606 functions question ending in a square root with two solutions to keep.

Solving a Modulus Function Equation

Solve |2x − 6| = 4 by splitting into two cases, and state the range of the modulus function, a 0606 question on |ax + b| the mark scheme wants seen in full.

Where a Function Meets Its Inverse

Find where a linear function equals its own inverse, and see why for an increasing function this reduces to solving f(x) = x, a 0606 functions question.