Question

Verify that these two are inverses using composition of functions and show your work

Answer 1

Answer:

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

Step-by-step explanation:

To prove two functions, say f(x) and g(x) are inverses to each other, we take the composition of the two functions: f(g(x)) and g(f(x)) and prove that:

                       f(g(x)) = g(f(x)) = x, x is the identity function

Now, given:

$f(x) = \frac{\sqrt[3]{x + 4}}{7}$         and          $g(x) = (7x)^3 - 4$

Calculate: f(g(x)):

$f(g(x)) = f((7x)^3 - 4)$

It means to substitute $(7x)^3 - 4$ in place of x.

Therefore, we get:

$f(g(x)) = \bigg \{ \frac{ \sqrt[3]{(7x)^3 - 4 + 4}}{7} \bigg \}$

$f(g(x)) = \bigg \{ \frac{ \sqrt[3]{(7x)^3}}{7} \bigg \}$

           $= \frac{(7x)}{7}$

           = x, the identity function.

Now compute g(f(x))

$g(x) = (7x)^3 - 4 \therefore g(x) = 343x^3 - 4$          

Now, $g(f(x)) = 343 \bigg \{ \bigg ( \frac{\sqrt[3]{x + 4}}{7} \bigg ) ^3 \bigg \} - 4$

                      $= 7^3 \frac{x + 4}{7^3} - 4$

                      $= (x + 4) - 4$

                      = x

We have proved f(g(x)) = g(f(x)) = x. Hence, they are inverses of each other.