To find the minimum and maximum values for the function [tex]\( k(x) = 10^x \)[/tex] within the given domain interval [tex]\([-3, 1]\)[/tex], we need to evaluate the function at the endpoints of the interval.
1. Identify the endpoints of the interval:
- The minimum value in the domain is [tex]\( x = -3 \)[/tex].
- The maximum value in the domain is [tex]\( x = 1 \)[/tex].
2. Evaluate the function at these points:
- For [tex]\( x = -3 \)[/tex]:
[tex]\[
k(-3) = 10^{-3} = \frac{1}{10^3} = \frac{1}{1000} = 0.001
\][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
k(1) = 10^1 = 10
\][/tex]
3. Determine the minimum and maximum values:
- The function attains its minimum value at [tex]\( x = -3 \)[/tex], which is
[tex]\[
k(-3) = 0.001
\][/tex]
- The function attains its maximum value at [tex]\( x = 1 \)[/tex], which is
[tex]\[
k(1) = 10
\][/tex]
Therefore, the minimum value of the function [tex]\( k(x) \)[/tex] over the given interval is [tex]\( 0.001 \)[/tex] and the maximum value is [tex]\( 10 \)[/tex].
Hence, the correct answer is:
- Minimum value [tex]\( = 0.001 \)[/tex]
- Maximum value [tex]\( = 10 \)[/tex]
So, the answer is:
[tex]\[
\boxed{\text{minimum value } = 0.001 \text{; maximum value } = 10}
\][/tex]
