C3 January 2011 Q6

EdexcelOld spec13 marksFunctions (including |mod|)

6. The function f is defined by

\[\mathrm{f} : x \mapsto \frac{3 - 2x}{x - 5}, \quad x \in \mathbb{R},\ x \ne 5\]

(a) Find \(\mathrm{f}^{-1}(x)\). (3)

Figure 2: graph of y = g(x), straight line from (-1, -9) through (0, -6) to (2, 0), then straight line from (2, 0) to (8, 4) — Figure 2

The function g has domain \(-1 \leqslant x \leqslant 8\), and is linear from \((-1, -9)\) to \((2, 0)\) and from \((2, 0)\) to \((8, 4)\). Figure 2 shows a sketch of the graph of \(y = \mathrm{g}(x)\).

(b) Write down the range of g. (1)

(d) Find \(\mathrm{fg}(8)\). (2)

(e) On separate diagrams, sketch the graph with equation

(i) \(y = \left|\mathrm{g}(x)\right|\),

(ii) \(y = \mathrm{g}^{-1}(x)\).

Show on each sketch the coordinates of each point at which the graph meets or cuts the axes. (4)

(f) State the domain of the inverse function \(\mathrm{g}^{-1}\). (1)

Mark scheme (a)

Scheme	Marks
\(y = \dfrac{3 - 2x}{x - 5} \Rightarrow y(x - 5) = 3 - 2x\)	M1
\(xy - 5y = 3 - 2x\)
\(\Rightarrow xy + 2x = 3 + 5y \Rightarrow x(y + 2) = 3 + 5y\)	M1
\(\Rightarrow x = \dfrac{3 + 5y}{y + 2} \qquad \therefore\ \mathrm{f}^{-1}(x) = \underline{\dfrac{3 + 5x}{x + 2}}\)	A1 oe
	(3)

Notes

M1: Attempt to make \(x\) (or swapped \(y\)) the subject

M1: Collect \(x\) terms together and factorise.

A1 oe: \(\underline{\dfrac{3 + 5x}{x + 2}}\)

Sketch of y = g^(-1)(x): line through (-6, 0) and (0, 2), then a steeper line from (0, 2) — Mark scheme (e)

small sketch of the required shape: shallow line then steeper line — Mark scheme (e)

Scheme	Marks
Range of g is \(-9 \leqslant \mathrm{g}(x) \leqslant 4\) or \(-9 \leqslant y \leqslant 4\)	B1
	(1)

Scheme	Marks
\(\mathrm{g}\,\mathrm{g}(2) = \mathrm{g}(0) = -6\), from sketch.	M1 A1
	(2)

Scheme	Marks
(e)(ii)	B1 B1
	(4)

Scheme	Marks
Domain of \(\mathrm{g}^{-1}\) is \(-9 \leqslant x \leqslant 4\)	B1ft
	(1)
	(13 marks)

Scheme	Marks
\(\mathrm{fg}(8) = \mathrm{f}(4)\)	M1
\(= \dfrac{3 - 4(2)}{4 - 5} = \dfrac{-5}{-1} = \underline{5}\)	A1
	(2)