A-Level Functions: worked solution

Question

Take f: x → 4/(x−4)² with domain restricted to x<4, so that f⁻¹ exists. Express f⁻¹ in a comparable form. [3]

Method: put y = f(x), make x the subject, then swap the letters. The domain of f⁻¹ is the range of f, so that has to be worked out too. Step 1: Write y = 4/(x - 4)², remembering the restriction x < 4. Step 2: Make the squared bracket the subject. y(x - 4)² = 4 (x - 4)² = 4/y Step 3: Take the square root. This gives two possibilities: x - 4 = 2/√y or x - 4 = -2/√y The domain is x < 4, so x - 4 is negative, and only the negative root can be correct. x - 4 = -2/√y x = 4 - 2/√y Step 4: Find the range of f, since that becomes the domain of f⁻¹. For x < 4 the bracket (x - 4) runs through every negative number, so (x - 4)² takes every value greater than 0 (never 0, since x ≠ 4). Hence 4/(x - 4)² takes every value greater than 0, and range of f = (0, ∞). So the domain of f⁻¹ is x > 0. (This also matches the algebra: √y needs y > 0.) Step 5: Rewrite with x as the input variable. f⁻¹: x → 4 - 2/√x, x ∈ R, x > 0. Check: f⁻¹(f(2)) with f(2) = 4/(2-4)² = 1 gives 4 - 2/√1 = 2. ✓