Verify that f(x) and g(x) are inverse functions: F(x)=3x^3+5, g(x)=3\sqrt[n]{x}x-5

Question

Verify that f(x) and g(x) are inverse functions: F(x)=3x^3+5, g(x)=3
`\sqrt[n]{x}`x-5

Answer 1

Shown Below

Explanation:

The question says:

`f(x)=3x^3+5`

`g(x)=\sqrt[3]{(x-5)/(3)}`

And it says to verify that both functions are inverses of each other.

To show this, we have to understand one composition of function property. When 2 functions are inverses of each other, the composition of both functions should yield "x". In notation:

(f o g)(x) = f(g(x)) = x

and

(g o f)(x) = g(f(x)) = x

So, we need to show that putting f(x) into g(x) and putting g(x) into f(x) both yields "x". Lets show this:

First:

`(fog)(x)=f(g(x))=3(\sqrt[3]{(x-5)/(3)} )^3+5=3((x-5)/(3))+5=x-5+5=x`

Verified.

Second:

`(gof)(x)=g(f(x))=\sqrt[3]{((3x^3+5)-5)/(3)}=\sqrt[3]{(3x^3)/(3)}=\sqrt[3]{x^3} =x`

Verified.

We have shown that both the functions are inverse of each other.