Question

Select the correct answer. Consider these functions: $($)=3.13 + 2 g(t)=V Which statements, if any, are true about these functions? 1. The function fg(x)) = x for all real x. II. The function & x)) = x for all real x. III. Functions fand gare inverse functions. O A. I only B. ll only C.I, II, and III D. None of the statements are true.

Answer 1

$\begin{gathered} f(x)=3x^3+2 \\ g(x)=\sqrt[3]{\frac{x-2}{3}} \end{gathered}$

Let's verify every statement:

I.

$f(g(x))=3(\sqrt[3]{\frac{x-2}{3}})^3+2=3(\frac{x-2}{3})+2=x-2+2=x$

So, the first one is true.

II:

$g(f(x))=\sqrt[3]{\frac{(3x^3+2)-2}{3}}=\sqrt[3]{x^3}=x$

So, this one is true

III.

Since:

$(fog)(x)=(gof)(x)=x$

We can conclude that f(x) and g(x) are inverse functions. So, this statements is also true

Answer:

C.I, II, and III