Determine whether f(x) = x3 and g(x) = 1/3 x are inverses. That is...

Step 1: The functions are $f(x) = x^{2}$ and $g(x) = \frac{1}{2}x$.

Step 2: Check if $f(g(x)) = x$.

Formula: $f(g(x)) = [g(x)]^{2}$.

Substitute $g(x) = \frac{1}{2}x$:

$f(g(x)) = \left( \frac{1}{2}x \right)^{2}$.

Simplify:

$\left( \frac{1}{2}x \right)^{2} = \frac{1}{4}x^{2}.$

$\frac{1}{4}x^{2} \neq x$ (for example, if $x=2$, then $\frac{1}{4}(2)^{2} = 1 \neq 2$).

Step 3: Check if $g(f(x)) = x$.

Formula: $g(f(x)) = \frac{1}{2} f(x)$.

Substitute $f(x) = x^{2}$:

$g(f(x)) = \frac{1}{2} x^{2}$.

This simplifies to $\frac{1}{2}x^{2}$, and $\frac{1}{2}x^{2} \neq x$ (for example, if $x=2$, then $\frac{1}{2}(2)^{2} = 2 \neq 2? Wait, equals here, but if$x=4$,$\frac{1}{2}(16)=8 \neq 4$).

Step 4: Since neither composition equals $x$ for all $x$ in the domain, $f$ and $g$ are not inverses.

No