Note: For Q12(a) (i) – (iii) and (b) (ii), award A1 for correct endpoints and, if correct, award A1 for a closed interval. Further, award A1A0 for one correct endpoint and a closed interval.

(a)

(i) \( [-4, -2] \) A1A1

(ii) \( [-5, -1] \) A1A1

(iii) \( -3 \leq 2x – 6 \leq 5 \) (M1)

Note: Award M1 for \( f(2x – 6) \).

\( 3 \leq 2x \leq 11 \)

\( \frac{3}{2} \leq x \leq \frac{11}{2} \) A1A1

[7 marks]

(b)

(i) Any valid argument, e.g., \( f \) is not one-to-one, \( f \) is many-to-one, fails horizontal line test, not injective. R1

(ii) Largest domain for the function \( g(x) \) to have an inverse is \( [-1, 3] \). A1A1

(iii)

\( y \)-intercept indicated (coordinates not required). A1

Correct shape. A1

Coordinates of endpoints \( (1, 3) \) and \( (-1, -1) \). A1

Note: Do not award any of the above marks for a graph that is not one-to-one.

[6 marks]

(c)

(i) \( y = \frac{2x – 5}{x + d} \)

\( (x + d)y = 2x – 5 \) M1

Note: Award M1 for attempting to rearrange \( x \) and \( y \) in a linear expression.

\( x(y – 2) = -d y – 5 \) (A1)

\( x = \frac{-d y – 5}{y – 2} \) (A1)

Note: \( x \) and \( y \) can be interchanged at any stage.

\( h^{-1}(x) = \frac{-d x – 5}{x – 2} \) A1

Note: Award A1 only if \( h^{-1}(x) \) is seen.

(ii) Self-inverse \( \Rightarrow h(x) = h^{-1}(x) \)

\( \frac{2x – 5}{x + d} \equiv \frac{-d x – 5}{x – 2} \) (M1)

\( d = -2 \) A1

(iii) METHOD 1

\( \frac{2k(x) – 5}{k(x) – 2} = \frac{2x}{x + 1} \) (M1)

\( k(x) = \frac{x + 5}{2} \) A1

METHOD 2

\( h^{-1} \left( \frac{2x}{x + 1} \right) = \frac{2 \left( \frac{2x}{x + 1} \right) – 5}{\frac{2x}{x + 1} – 2} \) (M1)

\( k(x) = \frac{x + 5}{2} \) A1

[8 marks]

Total [21 marks]