Notation, vocabulary and manipulation: interpret the reverse process as the ‘inverse function’ interpret the succession of two functions as a ‘composite function’ (Higher tier only)

The idea

At Higher tier, you need to understand inverse and composite functions. The inverse function reverses the original function: if f(x) = 3x + 2, then f⁻¹(x) = (x − 2) ÷ 3, because applying f then f⁻¹ returns the original input. To find an inverse, write y = f(x), rearrange to make x the subject, then replace y with x. A composite function applies one function after another: fg(x) means apply g first, then f. For example, if f(x) = x + 1 and g(x) = 2x, then fg(3) = f(g(3)) = f(6) = 7. The order matters: fg(x) is not usually the same as gf(x). You should be able to interpret the reverse process as an inverse function and a succession of functions as a composite function.

What earns the marks

• Find the inverse of a simple linear function by rearranging and swapping the variable.
• Use correct notation for inverse functions, such as f⁻¹(x).
• Process a composite function such as fg(x) by applying g first and then f.
• Recognise that fg(x) means f(g(x)) and that order matters.
• Show that applying a function and its inverse in either order returns the original input.

Common mistakes and how to put them right

• Misunderstanding: thinking f⁻¹(x) means 1 ÷ f(x).

  Put it right: f⁻¹(x) is the inverse function that reverses the effect of f.

• Misunderstanding: applying f before g in fg(x).

  Put it right: Fg(x) means apply g first, then f.

• Misunderstanding: assuming fg(x) always equals gf(x).

  Put it right: Composite functions are not generally commutative; check the order.

• Misunderstanding: forgetting to swap x and y when finding an inverse.

  Put it right: After rearranging, replace y with x to express the inverse in terms of x.

Exam tips

• For inverse functions, show your rearrangement steps clearly so the method is visible.
• For composite functions, write out the inner function first, then substitute into the outer function.
• Check your answer by substituting a simple value into the original and composite functions to see if the results match.