Absolutely, let's evaluate the function [tex]\( h(x) = e^{-3x} \)[/tex] at the specified values and round the answers to three decimal places.
### Step-by-Step Solution:
1. Evaluating [tex]\( h\left(\frac{1}{2}\right) \)[/tex]:
[tex]\[
h\left(\frac{1}{2}\right) = e^{-3 \left(\frac{1}{2}\right)} = e^{-1.5}
\][/tex]
Using a calculator to find [tex]\( e^{-1.5} \)[/tex], we get:
[tex]\[
e^{-1.5} \approx 0.223
\][/tex]
Therefore, [tex]\( h\left(\frac{1}{2}\right) \approx 0.223 \)[/tex].
2. Evaluating [tex]\( h(2.5) \)[/tex]:
[tex]\[
h(2.5) = e^{-3(2.5)} = e^{-7.5}
\][/tex]
Using a calculator to find [tex]\( e^{-7.5} \)[/tex], we get:
[tex]\[
e^{-7.5} \approx 0.001
\][/tex]
Therefore, [tex]\( h(2.5) \approx 0.001 \)[/tex].
3. Evaluating [tex]\( h(-1) \)[/tex]:
[tex]\[
h(-1) = e^{-3(-1)} = e^{3}
\][/tex]
Using a calculator to find [tex]\( e^{3} \)[/tex], we get:
[tex]\[
e^{3} \approx 20.086
\][/tex]
Therefore, [tex]\( h(-1) \approx 20.086 \)[/tex].
4. Evaluating [tex]\( h(-\pi) \)[/tex]:
[tex]\[
h(-\pi) = e^{-3(-\pi)} = e^{3\pi}
\][/tex]
Using a calculator to find [tex]\( e^{3\pi} \)[/tex], we get:
[tex]\[
e^{3\pi} \approx 12391.648
\][/tex]
Therefore, [tex]\( h(-\pi) \approx 12391.648 \)[/tex].
### Summary:
[tex]\[
\begin{array}{ll}
h(x) = e^{-3x} & \\
h\left(\frac{1}{2}\right) \approx 0.223 & \\
h(2.5) \approx 0.001 & \\
h(-1) \approx 20.086 & \\
h(-\pi) \approx 12391.648 & \\
\end{array}
\][/tex]
These are the rounded values of the function [tex]\( h(x) \)[/tex] at the given points.
