To find the domain and range of [tex]\( g(x) = -f\left(\frac{1}{4}(x+5)\right) \)[/tex], we will need to consider how the transformations inside the function [tex]\( g(x) \)[/tex] affect the domain and range of the original function [tex]\( f(x) \)[/tex].
### Finding the Domain of [tex]\( g(x) \)[/tex]:
1. Original Domain of [tex]\( f(x) \)[/tex]: The domain of [tex]\( f(x) \)[/tex] is given as [tex]\([-1, 4]\)[/tex].
2. Transformation Inside [tex]\( g(x) \)[/tex]: Within [tex]\( g(x) \)[/tex], the argument of [tex]\( f \)[/tex] is transformed to [tex]\( \frac{1}{4}(x + 5) \)[/tex].
3. Set Up the Inequality:
We need [tex]\( \frac{1}{4}(x + 5) \)[/tex] to lie within the domain [tex]\([-1, 4]\)[/tex]. Therefore, we set up the inequality:
[tex]\[
-1 \leq \frac{1}{4}(x + 5) \leq 4
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
- Start with the left part of the inequality:
[tex]\[
-1 \leq \frac{1}{4}(x + 5)
\][/tex]
Multiply all parts by 4:
[tex]\[
-4 \leq x + 5
\][/tex]
Subtract 5 from all parts:
[tex]\[
-9 \leq x
\][/tex]
- Now, solve the right part of the inequality:
[tex]\[
\frac{1}{4}(x + 5) \leq 4
\][/tex]
Multiply all parts by 4:
[tex]\[
x + 5 \leq 16
\][/tex]
Subtract 5 from all parts:
[tex]\[
x \leq 11
\][/tex]
- Combining both parts, the domain of [tex]\( g(x) \)[/tex] is:
[tex]\[
-9 \leq x \leq 11
\][/tex]
- In interval notation, the domain is:
[tex]\[
\boxed{[-9, 11]}
\][/tex]
### Finding the Range of [tex]\( g(x) \)[/tex]:
1. Original Range of [tex]\( f(x) \)[/tex]: The range of [tex]\( f(x) \)[/tex] is given as [tex]\([6, 7]\)[/tex].
2. Transformation in [tex]\( g(x) \)[/tex]:
- The function [tex]\( g(x) \)[/tex] outside transformation is [tex]\( -f(...) \)[/tex].
- This means each value of [tex]\( f(x) \)[/tex] is negated.
3. Negate the Range of [tex]\( f(x) \)[/tex]:
- If the values of [tex]\( f(x) \)[/tex] range from 6 to 7, then negating these values will result in:
[tex]\[
[-7, -6]
\][/tex]
4. In interval notation, the range of [tex]\( g(x) \)[/tex] is:
[tex]\[
\boxed{[-7, -6]}
\][/tex]
Combining both components, the final answer is:
- Domain: [tex]\([-9, 11]\)[/tex]
- Range: [tex]\([-7, -6]\)[/tex]
