To evaluate the function \( g(x) \) at the given values, we need to substitute each value of \( x \) into the function \( g(x) = \left(\frac{6}{5}\right)^{5x} \) and calculate the result.
1. Evaluate \( g\left(-\frac{1}{2}\right) \):
[tex]\[
g\left(-\frac{1}{2}\right) = \left(\frac{6}{5}\right)^{5 \left(-\frac{1}{2}\right)} = \left(\frac{6}{5}\right)^{-2.5}
\][/tex]
Using a calculator, we get:
[tex]\[
g\left(-\frac{1}{2}\right) \approx 0.633938145260609
\][/tex]
2. Evaluate \( g(\sqrt{3}) \):
[tex]\[
g(\sqrt{3}) = \left(\frac{6}{5}\right)^{5 \sqrt{3}}
\][/tex]
Using a calculator, we get:
[tex]\[
g(\sqrt{3}) \approx 4.84986562514995
\][/tex]
3. Evaluate \( g(-3) \):
[tex]\[
g(-3) = \left(\frac{6}{5}\right)^{-15}
\][/tex]
Using a calculator, we get:
[tex]\[
g(-3) \approx 0.0649054715188745
\][/tex]
4. Evaluate \( g\left(\frac{6}{5}\right) \):
[tex]\[
g\left(\frac{6}{5}\right) = \left(\frac{6}{5}\right)^6
\][/tex]
Using a calculator, we get:
[tex]\[
g\left(\frac{6}{5}\right) \approx 2.9859839999999993
\][/tex]
Thus, the evaluated values for the given function at the specific points are:
[tex]\[
g\left(-\frac{1}{2}\right) \approx 0.633938145260609
\][/tex]
[tex]\[
g(\sqrt{3}) \approx 4.84986562514995
\][/tex]
[tex]\[
g(-3) \approx 0.0649054715188745
\][/tex]
[tex]\[
g\left(\frac{6}{5}\right) \approx 2.9859839999999993
\][/tex]
