Question

Prove that () = + 5 and () = √ − 5 are inverse functions without finding the inverse of either function.

Answer 1

if two functions are inverse of each other, that means that their composite will result in "x" alone.

meaning, that if f(x) and g(x) are inverse of each other, then f( g(x) ) = x and the other way around will be the same, hmmm let's see if that's true

`f(x)=x^3 + 5\hspace{5em}g(x)=\sqrt[3]{x-5} \\\\[-0.35em] ~\dotfill\\\\ f( ~~ g(x) ~~ )=[g(x)]^3 + 5\implies f( ~~ g(x) ~~ )=[\sqrt[3]{x-5}]^3 + 5 \\\\\\ f( ~~ g(x) ~~ )=(x-5) + 5\implies f( ~~ g(x) ~~ )=x ~~ \textit{\LARGE \checkmark} \\\\[-0.35em] ~\dotfill\\\\ g( ~~ f(x) ~~ )=\sqrt[3]{[f(x)]-5}\implies g( ~~ f(x) ~~ )=\sqrt[3]{[x^3 + 5]-5} \\\\\\ g( ~~ f(x) ~~ )=\sqrt[3]{x^3}\implies g( ~~ f(x) ~~ )=x ~~ \textit{\LARGE \checkmark}`