CAIE A-Level Mathematics 1.2.3 One-to-One and Inverse Functions

CAIE A-Level Mathematics 1.2.3 One-to-One and Inverse Functions
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise deciding whether a function is one-to-one, rearranging y = f(x) to find its inverse and stating the inverse domain and range with any branch restrictions.

How this is tested

  • justify that an inverse exists using one-to-one behaviour or the horizontal-line test
  • swap variables and rearrange y = f(x), choosing the branch fixed by the original domain
  • state the inverse domain from the original range and retain excluded values or endpoints

Question 6(b)

[Maximum number: 2]

Functions f and g are defined by

f(x)=(x+3)2−12 for x⩾0, g(x)=2x−5 for x∈R.\begin{array}{ll} \mathrm{f}(x)=(x+3)^{2}-12 & \text { for } x \geqslant 0, \\ \mathrm{~g}(x)=2 x-5 & \text { for } x \in \mathbb{R} . \end{array}

Find an expression for f−1(x)\mathrm{f}^{-1}(x).