To solve the equation [tex]\(\operatorname{gh}(x) = 31\)[/tex] where [tex]\( g(x) = 4 - x \)[/tex] and [tex]\( h(x) = x^3 \)[/tex], we need to interpret [tex]\(\operatorname{gh}(x)\)[/tex] as [tex]\(g(h(x))\)[/tex].
1. First, we compute [tex]\(h(x)\)[/tex]:
[tex]\[
h(x) = x^3
\][/tex]
2. Next, we compute [tex]\(g(h(x))\)[/tex]:
[tex]\[
g(h(x)) = g(x^3)
\][/tex]
3. Substitute [tex]\(x^3\)[/tex] into the function [tex]\(g(x)\)[/tex]:
[tex]\[
g(x^3) = 4 - x^3
\][/tex]
4. Set [tex]\(g(x^3)\)[/tex] equal to 31:
[tex]\[
4 - x^3 = 31
\][/tex]
5. Solve the equation:
[tex]\[
4 - x^3 = 31
\][/tex]
6. Isolate [tex]\(x^3\)[/tex] by subtracting 4 from both sides:
[tex]\[
-x^3 = 31 - 4
\][/tex]
[tex]\[
-x^3 = 27
\][/tex]
7. Multiply both sides by -1 to solve for [tex]\(x^3\)[/tex]:
[tex]\[
x^3 = -27
\][/tex]
8. Take the cube root of both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \sqrt[3]{-27}
\][/tex]
[tex]\[
x = -3
\][/tex]
Thus, the solution to the equation [tex]\(\operatorname{gh}(x) = 31\)[/tex] is:
[tex]\[
x = -3
\][/tex]
