Question

confirm that F and g are inverses by showing that f(g(x))=x and g(f(x))=x f(x)=x^3+4 and g(x)=3(sqrt of x-4)

Answer 1

we are given

$f(x)=x^3+4$

$g(x)=\sqrt[3]{x-4}$

Calculation of f(g(x)):

$f(x)=x^3+4$

replace x = g(x)

$f(g(x))=(g(x))^3+4$

we can plug g(x)

$f(g(x))=(\sqrt[3]{x-4})^3+4$

$f(g(x))=x-4+4$

$f(g(x))=x$

Calculation of g(f(x)):

we are given

$g(x)=\sqrt[3]{x-4}$

replace x =f(x)

$g(f(x))=\sqrt[3]{f(x)-4}$

we can replace f(x)

we get

$g(f(x))=\sqrt[3]{x^3+4-4}$

$g(f(x))=\sqrt[3]{x^3}$

$g(f(x))=x$

we can see that

$f(g(x))=g(f(x))=x$

so, f(x) and g(x) are inverse of each other..........Answer