To find the pair of functions that are inverses of each other, we should check through the options.
Option A
[tex]\begin{gathered} f(x)\text{ = }\frac{x}{5}\text{ + 6} \\ g(x)\text{ = 5x -6} \end{gathered}[/tex]First, we set f(x) = y. Thus:
[tex]y=\text{ }\frac{x}{5}\text{ + 6}[/tex]Then, we swap the variables:
[tex]x\text{ = }\frac{y}{5}\text{ + 6}[/tex]Make y the subject of formula:
[tex]\begin{gathered} x\text{ = }\frac{y}{5}\text{ + 6} \\ \frac{y}{5}\text{ = x -6} \\ y\text{ =5x -30} \end{gathered}[/tex]But:
[tex]g(x)\text{ }\ne\text{ 5x -30}[/tex]Hence, option A is incorrect
Option B
[tex]\begin{gathered} f(x)\text{ =}\frac{\sqrt[3]{x}}{6} \\ g(x)=6x^3 \end{gathered}[/tex]Set f(x) = y. Thus:
[tex]y\text{ = }\frac{\sqrt[3]{x}}{6}[/tex]Swap the variables:
[tex]x\text{ = }\frac{\sqrt[3]{y}}{6}[/tex]Make y the subject of the formula:
[tex]\begin{gathered} 6x\text{ = }\sqrt[3]{y} \\ \text{Cube both sides} \\ (6x)^3\text{ = y} \\ y\text{ = }216x^3 \end{gathered}[/tex]But:
[tex]g(x)\text{ }\ne216x^3[/tex]Hence, Option B is incorrect
Option C:
[tex]\begin{gathered} f(x)\text{ = }7x\text{ -2} \\ g(x)\text{ = }\frac{x\text{ + 2}}{7} \end{gathered}[/tex]Set f(x) = y. Thus:
[tex]y\text{ = 7x - 2}[/tex]Swap the variable:
[tex]x\text{ = 7y -2}[/tex]Make y the subject of formula:
[tex]\begin{gathered} x\text{ + 2 = 7y} \\ 7y\text{ = x +2} \\ \text{Divide both sides by 7} \\ \frac{7y}{7}\text{ = }\frac{x+2}{7} \\ y\text{ = }\frac{x+2}{7} \end{gathered}[/tex]But :
[tex]g(x)\text{ = }\frac{x+2}{7}[/tex]Hence, option C is correct
Option D
[tex]\begin{gathered} f(x)\text{ = }\frac{5}{x}\text{ -2} \\ g(x)\text{ =}\frac{x+2}{5} \end{gathered}[/tex]Set y =f(x). Thus:
[tex]y\text{ = }\frac{5}{x}\text{ -2}[/tex]Swap the variables:
[tex]x\text{ = }\frac{5}{y}\text{ -2}[/tex]Make y the subject of formula:
[tex]undefined[/tex]