EXN.1.SL.TZ0.5

pestleMathematicsAASLPaper 1EXN· sl-2-5-composite-functions-identity-finding-inversesource ↗

The functions $f$ and $g$ are defined for $x \in ℝ$ by $f (x) = x - 2$ and $g (x) = a x + b$ , where $a, b \in ℝ$ .

Given that $(f \circ g) (2) = - 3$ and $(g \circ f) (1) = 5$ , find the value of $a$ and the value of $b$ .

Markscheme / solution

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

$(f \circ g) (x) = a x + b - 2$ (M1)

$(f \circ g) (2) = - 3 \Rightarrow 2 a + b - 2 = - 3 (2 a + b = - 1)$ A1

$(g \circ f) (x) = a (x - 2) + b$ (M1)

$(g \circ f) (1) = 5 \Rightarrow - a + b = 5$ A1

a valid attempt to solve their two linear equations for $a$ and $b$ M1

so $a = - 2$ and $b = 3$ A1

[6 marks]

Examiners’ report

[N/A]