To solve the equation [tex]\( g(h(x)) = 31 \)[/tex] given the functions [tex]\( g(x) = 4 - x \)[/tex] and [tex]\( h(x) = x^3 \)[/tex], we can follow these steps:
1. Express the composite function [tex]\( g(h(x)) \)[/tex]:
- We start by finding [tex]\( h(x) \)[/tex], which is given as [tex]\( h(x) = x^3 \)[/tex].
- Next, substitute [tex]\( h(x) \)[/tex] into [tex]\( g(x) \)[/tex], i.e., [tex]\( g(h(x)) \)[/tex].
- Since [tex]\( g(x) = 4 - x \)[/tex], we have [tex]\( g(h(x)) = g(x^3) = 4 - x^3 \)[/tex].
2. Set up the equation given in the problem:
- We are given that [tex]\( g(h(x)) = 31 \)[/tex].
- Substitute the composite function into the equation: [tex]\( 4 - x^3 = 31 \)[/tex].
3. Solve for [tex]\( x \)[/tex]:
- Start by isolating the cubic term:
[tex]\[
4 - x^3 = 31
\][/tex]
- Subtract 4 from both sides to simplify:
[tex]\[
-x^3 = 31 - 4
\][/tex]
[tex]\[
-x^3 = 27
\][/tex]
- Divide both sides by -1 to get:
[tex]\[
x^3 = -27
\][/tex]
- Solve for [tex]\( x \)[/tex] by taking the cube root of both sides:
[tex]\[
x = \sqrt[3]{-27}
\][/tex]
4. Calculate the cube root of -27:
- The cube root of -27 is:
[tex]\[
x = (-27)^{\frac{1}{3}}
\][/tex]
Hence, the solution to the equation [tex]\( g(h(x)) = 31 \)[/tex] is:
[tex]\[
x = 1.5 + 2.598j
\][/tex]
