Question

Verify that f(x) and g(x) are inverse functions: F(x)=3x^3+5, g(x)=3"TexFormula1" title="\sqrt[n]{x}" alt="\sqrt[n]{x}" align="absmiddle" class="latex-formula">x-5

Answer 1

Answer:

Shown Below

Step-by-step explanation:

The question says:

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

$g(x)=\sqrt[3]{\frac{x-5}{3}}$

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:

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

Verified.

Second:

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

Verified.

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