Answer
$g(-\frac{1}{2})\approx 0.650$;
$g(\sqrt6) \approx 8.281$;
$g(-3)\approx 0.075$;
$g(\frac{4}{3}) \approx 3.160$
Work Step by Step
Evaluate the function for the given values of $x$ by substituting each $x$ into $g(x)$ then using a calculator to find the exact value.
When x=$-\frac{1}{2}$:
$g(x) = \left(\dfrac{4}{3}\right)^{3x}
\\g(-\frac{1}{2})=\left(\dfrac{4}{3}\right)^{3(-\frac{1}{2})}
\\g(-\frac{1}{2})=\left(\dfrac{4}{3}\right)^{-\frac{3}{2}}
\\g(-\frac{1}{2})\approx 0.650$
When $x=\sqrt6$:
$g(x) = \left(\dfrac{4}{3}\right)^{3x}
\\g(\sqrt6) = \left(\dfrac{4}{3}\right)^{3(\sqrt6)}
\\g(\sqrt6) = \left(\dfrac{4}{3}\right)^{3\sqrt6}
\\g(\sqrt6) \approx 8.281$
When $x=-3$:
$g(x)=\left(\dfrac{4}{3}\right)^{3x}
\\g(-3)=\left(\dfrac{4}{3}\right)^{3(-3)}
\\g(-3)=\left(\dfrac{4}{3}\right)^{-9}
\\g(-3)\approx 0.075$
When $x=\frac{4}{3}$:
$g(x)=\left(\dfrac{4}{3}\right)^{3x}
\\g(\frac{4}{3}) = \left(\dfrac{4}{3}\right)^{3(\frac{4}{3})}
\\g(\frac{4}{3}) = \left(\dfrac{4}{3}\right)^{4}
\\g(\frac{4}{3}) \approx 3.160$
