21N.1.AHL.TZ0.2

pestleMathematicsAAHLPaper 121N· sl-2-2-functions-notation-domain-range-and-inverse-as-reflectionsource ↗

The function $f$ is defined by $f (x) = \frac{2 x + 4}{3 - x}$ , where $x \in ℝ, x \neq 3$ .

Write down the equation of

Find the coordinates where the graph of $f$ crosses

the vertical asymptote of the graph of $f$ .

[1]

a.i.

the horizontal asymptote of the graph of $f$ .

[1]

a.ii.

the $x$ -axis.

[1]

b.i.

the $y$ -axis.

[1]

b.ii.

Sketch the graph of $f$ on the axes below.

[1]

The function $g$ is defined by $g (x) = \frac{a x + 4}{3 - x}$ , where $x \in ℝ, x \neq 3$ and $a \in ℝ$ .

Given that $g (x) = g^{- 1} (x)$ , determine the value of $a$ .

[4]

Markscheme / solution

$x = 3$ A1

[1 mark]

a.i.

$y = - 2$ A1

[1 mark]

a.ii.

$(- 2, 0)$ (accept $x = - 2$ ) A1

[1 mark]

b.i.

$(0, \frac{4}{3})$ (accept $y = \frac{4}{3}$ and $f (0) = \frac{4}{3}$ ) A1

[1 mark]

b.ii.

Note: Award A1 for completely correct shape: two branches in correct quadrants with asymptotic behaviour.

[1 mark]

METHOD 1

$(g (x) =) y = \frac{a x + 4}{3 - x}$

attempt to find $x$ in terms of $y$ (M1)

OR exchange $x$ and $y$ and attempt to find $y$ in terms of $x$

$3 y - x y = a x + 4$ A1

$a x + x y = 3 y - 4$

$x (a + y) = 3 y - 4$

$x = \frac{3 y - 4}{y + a}$

$g^{- 1} (x) = \frac{3 x - 4}{x + a}$ A1

Note: Condone use of $y =$

$g (x) \equiv g^{- 1} (x)$

$\frac{a x + 4}{3 - x} \equiv \frac{3 x - 4}{x + a}$

$\Rightarrow a = - 3$ A1

METHOD 2

$g (x) = \frac{a x + 4}{3 - x}$

attempt to find an expression for $g (g (x))$ and equate to $x$ (M1)

$g g (x) = \frac{a (\frac{a x + 4}{3 - x}) + 4}{3 - (\frac{a x + 4}{3 - x})} = x$ A1

$\frac{a (a x + 4) + 4 (3 - x)}{(9 - 3 x) - (a x + 4)} = x$

$\frac{a (a x + 4) + 4 (3 - x)}{5 - (3 + a) x} = x$

$a (a x + 4) + 4 (3 - x) = x (5 - (3 + a) x)$ A1

equating coefficients of $x^{2}$ (or similar)

$a = - 3$ A1

[4 marks]

Examiners’ report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]