WAEC 2017 · Paper 2 · Q1

(a)
If $f(x) = \dfrac{4 - 5x}{2}$ and $g(x) = x + 6$ , $x \in \mathbb R$ , find $f \circ g^{-1}$ .
(b)
$P(x, y)$ divides the line joining $(7, -5)$ and $(-2, 7)$ internally in the ratio $5 : 4$ . Find the coordinates of $P$ .
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

(a)

Find $g^{-1}$ first: $y = x + 6$ gives $x = y - 6$ , so $g^{-1}(x) = x - 6$ .
$g^{-1}$ acts first: $f \circ g^{-1}(x) = f(x - 6)$
$= \dfrac{4 - 5(x - 6)}{2}$ .
Multiply out the top: $4 - 5x + 30 = 34 - 5x$ .
Divide each term by 2: $f \circ g^{-1}(x) = 17 - \frac52x$ .

(b)

$P$ divides $A(7, -5)$ to $B(-2, 7)$ in the ratio $5 : 4$ , so $P = \left(\dfrac{4x_A + 5x_B}{9}, \dfrac{4y_A + 5y_B}{9}\right)$ .
The $x$ -coordinate: $\dfrac{4(7) + 5(-2)}{9} = \dfrac{18}{9}$ , which is 2.
The $y$ -coordinate: $\dfrac{4(-5) + 5(7)}{9} = \dfrac{15}{9}$
$= \dfrac53$ .
So $P\left(2, \frac53\right)$ .

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