Question(ii) Find $p(x)$ . $f(x)$ is defined by $f(x)=x^{3}+5, x \in \mathbb{R}$ . Find the value of $x$ such that $f g(2 x)=13$ if $g(x)=3 x-10$ . [3 marks]

Studdy Solution

STEP 1

1. The function $f(x)$ is defined as $f(x) = x^3 + 5$ .
2. The function $g(x)$ is defined as $g(x) = 3x - 10$ .
3. We need to find the value of $x$ such that $f(g(2x)) = 13$ .

STEP 2

1. Substitute $2x$ into the function $g(x)$ .
2. Substitute the result from step 1 into the function $f(x)$ .
3. Set the equation from step 2 equal to 13 and solve for $x$ .

STEP 3

Substitute $2x$ into the function $g(x)$ .
$g(2x) = 3(2x) - 10$ $g(2x) = 6x - 10$

STEP 4

Substitute the result from step 1 into the function $f(x)$ .
$f(g(2x)) = f(6x - 10)$ $f(6x - 10) = (6x - 10)^3 + 5$

STEP 5

Set the equation from step 2 equal to 13 and solve for $x$ .
$(6x - 10)^3 + 5 = 13$
Subtract 5 from both sides:
$(6x - 10)^3 = 8$
Take the cube root of both sides:
$6x - 10 = 2$
Add 10 to both sides:
$6x = 12$
Divide both sides by 6:
$x = 2$
The value of $x$ that satisfies the equation is:
$\boxed{2}$

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