Answer :
To solve the equation [tex]\( \operatorname{gh}(x) = 31 \)[/tex], we first need to understand what [tex]\(\operatorname{gh}(x)\)[/tex] represents.
Given the functions:
[tex]\[ g(x) = 4 - x \][/tex]
[tex]\[ h(x) = x^3 \][/tex]
The composite function [tex]\(\operatorname{gh}(x)\)[/tex] is defined as:
[tex]\[ \operatorname{gh}(x) = g(x) \cdot h(x) \][/tex]
We can now express [tex]\(\operatorname{gh}(x)\)[/tex] as:
[tex]\[ \operatorname{gh}(x) = (4 - x) \cdot x^3 \][/tex]
We set this equal to 31:
[tex]\[ (4 - x) \cdot x^3 = 31 \][/tex]
To solve for [tex]\( x \)[/tex], we rearrange the equation:
[tex]\[ (4 - x) \cdot x^3 - 31 = 0 \][/tex]
This is a polynomial equation in terms of [tex]\( x \)[/tex]. The correct numerical roots for this equation, obtained through symbolic calculation and simplification, are:
1. [tex]\( x = 1 - \frac{\sqrt{4 + \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}} + 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}}{2} - \frac{\sqrt{8 - 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3} - \frac{16}{\sqrt{4 + \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}} + 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}} - \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}}}{2} \)[/tex]
2. [tex]\( x = 1 + \frac{\sqrt{4 + \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}} + 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}}{2} - \frac{\sqrt{8 - 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3} + \frac{16}{\sqrt{4 + \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}} + 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}} - \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}}}{2} \)[/tex]
3. [tex]\( x = 1 + \frac{\sqrt{8 - 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3} - \frac{16}{\sqrt{4 + \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}} + 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}} - \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}}}{2} - \frac{\sqrt{4 + \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}} + 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}}{2} \)[/tex]
4. [tex]\( x = 1 + \frac{\sqrt{8 - 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3} + \frac{16}{\sqrt{4 + \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}} + 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}} - \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}}}{2} + \frac{\sqrt{4 + \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}} + 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}}{2} \)[/tex]
Therefore, the solutions to the equation are:
[tex]\[ x = \ldots \][/tex]
Given the functions:
[tex]\[ g(x) = 4 - x \][/tex]
[tex]\[ h(x) = x^3 \][/tex]
The composite function [tex]\(\operatorname{gh}(x)\)[/tex] is defined as:
[tex]\[ \operatorname{gh}(x) = g(x) \cdot h(x) \][/tex]
We can now express [tex]\(\operatorname{gh}(x)\)[/tex] as:
[tex]\[ \operatorname{gh}(x) = (4 - x) \cdot x^3 \][/tex]
We set this equal to 31:
[tex]\[ (4 - x) \cdot x^3 = 31 \][/tex]
To solve for [tex]\( x \)[/tex], we rearrange the equation:
[tex]\[ (4 - x) \cdot x^3 - 31 = 0 \][/tex]
This is a polynomial equation in terms of [tex]\( x \)[/tex]. The correct numerical roots for this equation, obtained through symbolic calculation and simplification, are:
1. [tex]\( x = 1 - \frac{\sqrt{4 + \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}} + 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}}{2} - \frac{\sqrt{8 - 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3} - \frac{16}{\sqrt{4 + \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}} + 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}} - \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}}}{2} \)[/tex]
2. [tex]\( x = 1 + \frac{\sqrt{4 + \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}} + 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}}{2} - \frac{\sqrt{8 - 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3} + \frac{16}{\sqrt{4 + \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}} + 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}} - \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}}}{2} \)[/tex]
3. [tex]\( x = 1 + \frac{\sqrt{8 - 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3} - \frac{16}{\sqrt{4 + \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}} + 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}} - \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}}}{2} - \frac{\sqrt{4 + \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}} + 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}}{2} \)[/tex]
4. [tex]\( x = 1 + \frac{\sqrt{8 - 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3} + \frac{16}{\sqrt{4 + \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}} + 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}} - \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}}}{2} + \frac{\sqrt{4 + \frac{62}{3 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}} + 2 \left( 31 + \frac{62 \sqrt{3} i}{9} \right)^{1/3}}}{2} \)[/tex]
Therefore, the solutions to the equation are:
[tex]\[ x = \ldots \][/tex]
