Let's start by understanding how the transformations will affect the domain and range of the function [tex]\( g(x) = 2f(5x) + 1 \)[/tex].
### Domain Transformation
1. Original Domain of [tex]\( f(x) \)[/tex]:
[tex]\[
[-20, 15]
\][/tex]
2. Adjusting for the inner function [tex]\( 5x \)[/tex] in [tex]\( f(5x) \)[/tex]:
[tex]\[
5x \text{ must lie within } [-20, 15]
\][/tex]
3. Solving for [tex]\( x \)[/tex] to find the new domain:
[tex]\[
-20 \leq 5x \leq 15
\][/tex]
[tex]\[
\frac{-20}{5} \leq x \leq \frac{15}{5}
\][/tex]
[tex]\[
-4 \leq x \leq 3
\][/tex]
4. Thus, the domain of [tex]\( g(x) \)[/tex] is:
[tex]\[
[-4, 3]
\][/tex]
### Range Transformation
1. Original Range of [tex]\( f(x) \)[/tex]:
[tex]\[
(-\infty, 5]
\][/tex]
2. Transforming [tex]\( f(x) \)[/tex] to [tex]\( 2f(5x) \)[/tex]:
- The multiplication by 2 scales the range by a factor of 2.
- Therefore, the new range is:
[tex]\[
2 \cdot (-\infty, 5] = (-\infty, 10]
\][/tex]
3. Addition of 1 in [tex]\( 2f(5x) + 1 \)[/tex]:
- Adding 1 shifts the entire range up by 1 unit.
- Therefore, the range now becomes:
[tex]\[
(-\infty, 10] + 1 = (-\infty, 11]
\][/tex]
4. Thus, the range of [tex]\( g(x) \)[/tex] is:
[tex]\[
(-\infty, 11]
\][/tex]
### Final Answer
- Domain: [tex]\(\boxed{[-4, 3]}\)[/tex]
- Range: [tex]\(\boxed{(-\infty, 11]}\)[/tex]
