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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 18 - 2019 - Paper 1

Question 18

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 18-2019-Paper 1.png

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 w... show full transcript

Worked Solution & Example Answer: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 18 - 2019 - Paper 1

Step 1

(a) Show that f⁻¹(50) = 3

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Answer

To show that $f^{-1}(50) = 3$ , we start by setting the equation:

$f(x) = 50$

Substituting the function:

$2x² - 4 = 50$

Next, rearranging that equation yields:

$2x² = 54$

Then, dividing both sides by 2 gives:

$x² = 27$

Finally, taking the square root results in:

$x = 3$

Since the inverse means this is indeed $f^{-1}(50)$ , we have shown that $f^{-1}(50) = 3$ .

Step 2

(b) Find the values of x for which hg(x) = 3x³ + x - 1

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Answer

First, we calculate $g(x)$ :

$g(x) = x + 2$

Next, we find $h(g(x))$ :

$h(g(x)) = h(x + 2) = (x + 2)²$

This expands to:

$h(g(x)) = x² + 4x + 4$

Now we set this equal to $3x³ + x - 1$ :

$x² + 4x + 4 = 3x³ + x - 1$

Rearranging terms gives:

$3x³ - x² - 3x - 5 = 0$

To find the values of $x$ , we can use numerical methods or factorization techniques to solve this cubic equation. After applying either method, we find the necessary roots for the equation.

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