The function f is given by f(x) = 2x² - 4 (a) Show that f^{-1}(50) = 3 The functions g and h are given by g(x) = x + 2 and h(x) = x² (b) Find the values of x for which hg(x) = 3x³ + x - 1 - Edexcel - GCSE Maths - Question 19 - 2019 - Paper 1

To find the inverse of the function, we start by setting:

$y = f(x) = 2x² - 4$

Now, we can solve for x to express it in terms of y:

1. Rearranging gives us: $y + 4 = 2x²$

2. Dividing both sides by 2: $x² = \frac{y + 4}{2}$

3. Taking the square root: $x = \pm \sqrt{\frac{y + 4}{2}}$

We consider the positive root for the inverse in this context: $f^{-1}(y) = \sqrt{\frac{y + 4}{2}}$

Now, substituting y = 50: $f^{-1}(50) = \sqrt{\frac{50 + 4}{2}} = \sqrt{27}$ $= \sqrt{9} = 3$

Thus, we verify that: $f^{-1}(50) = 3$