21M.1.AHL.TZ2.2

pestleMathematicsAIHLPaper 121M· sl-2-2-functions-notation-domain-range-and-inverse-as-reflectionsource ↗

A function is defined by $f (x) = 2 - \frac{12}{x + 5}$ for $- 7 \leq x \leq 7, x \neq - 5$ .

Find the range of $f$ .

[3]

Find an expression for the inverse function $f^{- 1} (x)$ . The domain is not required.

[3]

Write down the range of $f^{- 1} (x)$ .

[1]

Markscheme / solution

$(f (- 7) =) 8$ and $(f (7) =) 1$ (A1)

range is $f (x) \leq 1, f (x) \geq 8$ A1A1

Note: Award at most A1A1A0 if strict inequalities are used.

[3 marks]

interchanging $x, y$ at any stage (A1)

$y = 2 - \frac{12}{x + 5}$

$\frac{12}{x + 5} = 2 - y$

$\frac{12}{2 - y} = x + 5$ (A1)

$\frac{12}{2 - y} - 5 = x$

$(f^{- 1} (x) =) \frac{12}{2 - x} - 5 (= \frac{2 + 5 x}{2 - x})$ A1

[3 marks]

range is $- 7 \leq f^{- 1} (x) \leq 7, f^{- 1} (x) \neq - 5$ A1

[1 mark]

Examiners’ report

[N/A]