fg(x) means f(g(x))
Write the expression for g(x) inside f(...)

$f (g (x)) = f (x^{2} + 4)$

Find $f (x^{2} + 4)$ by replacing the x's in f(x) with the brackets $(x^{2} + 4)$

$\begin{array}{rcl} f (g (x)) & = & 3 (x^{2} + 4) - 1 \end{array}$

[1]

gf(x) means g(f(x))
Write the expression for f(x) inside g(...)

$g (f (x)) = g (3 x - 1)$

Find $g (3 x - 1)$ by replacing the x's in g(x) with the brackets $(3 x - 1)$

$g (f (x)) = {(3 x - 1)}^{2} + 4$

[1]

Set fg(x) equal to 2 × gf(x)

$3 (x^{2} + 4) - 1 = 2 [{(3 x - 1)}^{2} + 4]$

[1]

Expand the curved brackets on either side

$3 x^{2} + 12 - 1 = 2 [(3 x - 1) (3 x - 1) + 4] 3 x^{2} + 11 = 2 [9 x^{2} - 6 x + 1 + 4] 3 x^{2} + 11 = 2 [9 x^{2} - 6 x + 5]$

(3x - 1)² expanded correctly [1]

Expand the square brackets (by multiplying all terms inside by 2]

$3 x^{2} + 11 = 18 x^{2} - 12 x + 10$

Make the left-hand side zero (by subtracting 3x² and 11 from both sides)

$0 = 15 x^{2} - 12 x - 1$

$15 x^{2} - 12 x - 1 = 0$ [1]

Functions (AQA GCSE Maths)

Exam Questions