The functions f and g are defined by f : x ↦ 1 - 2x², x ∈ ℝ g : x ↦ 3 / (x - 4), x > 0, x ∈ ℝ (a) Find the inverse function fᶦ - Edexcel - A-Level Maths Pure - Question 2 - 2007 - Paper 5

To find the composite function gf, we will substitute g(x) into f:

First, we find g(x):

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

Now substituting g(x) into f:

$gf(x) = f(g(x)) = f \left( \frac{3}{x - 4} \right) = 1 - 2 \left( \frac{3}{x - 4} \right)²$

Calculating the expression:

= \frac{(x - 4)² - 18}{(x - 4)²} $$ Simplifying the numerator: $$ (x - 4)² - 18 = x² - 8x + 16 - 18 = x² - 8x - 2 $$ Finally, the desired form: $$ gf(x) = \frac{8x² - 1}{1 - 2x²} $$.

To solve for gf(x) = 0:

$gf(x) = \frac{8x² - 1}{1 - 2x²} = 0$

This implies that the numerator must be zero:

$8x² - 1 = 0$

Solving this gives:

x² = \frac{1}{8} \\ x = \pm \frac{1}{2\sqrt{2}} = \frac{1}{2 \sqrt{2}} Because g(x) is only defined for x > 0, we take: $$ x = \frac{1}{2 \sqrt{2}}. $$

To find the stationary point, we first compute the derivative of gf(x):

$$\frac{dy}{dx} = \frac{(1 - 2x²)(16x) - (8x² - 1)(4x)}{(1 - 2x²)²}$$

Setting the numerator to zero:

$(1 - 2x²)(16x) - (8x² - 1)(4x) = 0$

This leads to:

20x - 64x³ = 0 \\ x(20 - 64x²) = 0 $$ Thus, either: $$ x = 0 $$ or solving $20 - 64x² = 0$: $$ 64x² = 20 \\ x² = \frac{5}{16} \\ x = \frac{ oot{5}}{4}. $$ To find y-coordinates for stationary points, substitute x back into gf: For $x = 0$: $$ gf(0) = \frac{8(0)² - 1}{1 - 2(0)²} = -1 $$ So, the coordinates of the stationary point are: $$(0, -1). $$