B: Algebra and functions: Understand and use composite functions; inverse functions and their graphs.

The idea

This topic covers combining functions and reversing them. A composite function applies one function to the output of another: (f ∘ g)(x) = f(g(x)), read as 'f of g of x'. Order matters: f(g(x)) is generally not the same as g(f(x)). The domain of f(g(x)) consists of x in the domain of g for which g(x) lies in the domain of f. An inverse function f⁻¹ reverses f: if f(a) = b then f⁻¹(b) = a. To find f⁻¹, write y = f(x), swap x and y, and rearrange for y. The graph of y = f⁻¹(x) is the reflection of y = f(x) in the line y = x. For f⁻¹ to exist as a function, f must be one-to-one on its domain; restricting the domain can achieve this. The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.

What earns the marks

• Forming composite functions: substitute the whole expression for the inner function into the outer function and simplify correctly.
• Evaluating composite functions at a given value: work from the inside out, applying the inner function first.
• Finding inverse functions algebraically: write y = f(x), interchange x and y, then rearrange to make y the subject; denote the result f⁻¹(x).
• Graphs of inverse functions: sketch y = f⁻¹(x) as the reflection of y = f(x) in y = x, and identify that points (a, b) on f correspond to (b, a) on f⁻¹.
• Domain and range: state the domain and range of composite and inverse functions, using the original function's domain and range appropriately.
• Existence of inverses: recognise that a one-to-one function has an inverse; if not one-to-one, restrict the domain to create an inverse.

Common mistakes and how to put them right

• Assuming f(g(x)) = g(f(x)).

  Put it right by composing in the stated order and simplifying carefully.

• Writing f⁻¹(x) as 1/f(x).

  Put it right by finding the inverse by swapping x and y and rearranging, not by taking the reciprocal.

• Forgetting to restrict the domain when a function is not one-to-one.

  Put it right by stating a suitable restricted domain so that the inverse is a function.

• Mixing up domain and range of f and f⁻¹.

  Put it right by remembering that the domain of f⁻¹ is the range of f, and vice versa.

Exam tips

• Show the substitution step clearly when forming a composite function, especially if the inner function is a fraction or contains a modulus.
• When finding an inverse, always swap x and y explicitly before rearranging; this makes your method clear.
• For graph questions, draw the line y = x as a guide and reflect key points to sketch the inverse accurately.