CAIE A-Level Mathematics AS 1.2.3 Inverse Functions

[Maximum number: 6]

The function f is defined by $\mathrm{f}(x)=10+6 x-x^{2}$ for $x \in \mathbb{R}$ .

Figure for Question CAIE A-Level Mathematics AS 1.2.3 One One and Inverse Functions Questions question 1 — CAIE A-Level Mathematics AS

The function g is defined by g(x)=4 x+k for $x \in \mathbb{R}$ where k is a constant.

It is given that the graph of $y=\mathrm{g}^{-1} \mathrm{f}(x)$ meets the graph of y=g(x) at a single point P.

Determine the coordinates of P.

$\mathrm{g}^{-1}(x)=\frac{1}{4}(x-k)$ $g^{-1} f(x)=\frac{1}{4}\left(10+6 x-x^{2}-k\right)=4 x+k$
OE
May use their completed square form for f(x).

Simplify the quadratic equation obtained from $\mathrm{g}^{-1} \mathrm{f}(x)=\mathrm{g}(x)$ provided k is
present and apply $b^{2}-4 a c=0$ to this quadratic equation

*M1

Expect $x^{2}+10 x-10+5 k=0$ .

Obtain 100-4(5 k-10)=0 and hence k=7
Use their k to form and solve a quadratic in x
Marking guidance:

Allow if their quadratic has two solutions.

(-5,-13) only
SC B1 if no method seen.

Alternative Method for first 4 marks

State $\mathrm{f}(x)=\operatorname{gg}(x)$ $\operatorname{gg}(x)=16 x+5 k$
Apply $b^{2}-4 a c=0$ to quadratic equation obtained from $\mathrm{f}(x)=\operatorname{gg}(x)$

(*M1)

Provided k is present.

100-4(5 k-10)=0 and hence k=7

CAIE A-Level Mathematics AS 1.2.3 One One and Inverse Functions Questions question 1