To determine the domain and range of the function [tex]\( g(x) = -f\left(\frac{1}{4}(x + 5)\right) \)[/tex], given that the domain of [tex]\( f(x) \)[/tex] is [tex]\([-1,4]\)[/tex] and the range is [tex]\([6,7]\)[/tex], we need to follow a series of transformations.
### Step-by-Step Solution:
#### 1. Finding the Domain of [tex]\( g(x) \)[/tex]
The domain of [tex]\( g(x) \)[/tex] is determined by the expression inside the function [tex]\( f \)[/tex], i.e., [tex]\( \frac{1}{4}(x + 5) \)[/tex]. We need this expression to fall within the original domain of [tex]\( f \)[/tex], which is [tex]\([-1, 4]\)[/tex].
So we set up the inequality:
[tex]\[
-1 \leq \frac{1}{4}(x + 5) \leq 4
\][/tex]
Next, solve these inequalities for [tex]\( x \)[/tex]:
First inequality:
[tex]\[
-1 \leq \frac{1}{4}(x + 5)
\][/tex]
Multiply both sides by 4:
[tex]\[
-4 \leq x + 5
\][/tex]
Subtract 5 from both sides:
[tex]\[
x \geq -9
\][/tex]
Second inequality:
[tex]\[
\frac{1}{4}(x + 5) \leq 4
\][/tex]
Multiply both sides by 4:
[tex]\[
x + 5 \leq 16
\][/tex]
Subtract 5 from both sides:
[tex]\[
x \leq 11
\][/tex]
So, the domain of [tex]\( g(x) \)[/tex] is:
[tex]\[
[-9, 11]
\][/tex]
#### 2. Finding the Range of [tex]\( g(x) \)[/tex]
To determine the range of [tex]\( g(x) \)[/tex], we start with the given range of [tex]\( f(x) \)[/tex], which is [tex]\([6, 7]\)[/tex].
Since [tex]\( g(x) = -f\left(\frac{1}{4}(x + 5)\right) \)[/tex], we are applying a vertical reflection (multiplying by -1).
To find the new range, we multiply the values of the range of [tex]\( f(x) \)[/tex] by -1:
- The maximum value in the range of [tex]\( f(x) \)[/tex] will become the minimum value in the range of [tex]\( g(x) \)[/tex].
- The minimum value in the range of [tex]\( f(x) \)[/tex] will become the maximum value in the range of [tex]\( g(x) \)[/tex].
Original range of [tex]\( f(x) \)[/tex]:
[tex]\[
[6, 7]
\][/tex]
After multiplying by -1:
[tex]\[
-7 \text{ (from 6)}, -6 \text{ (from 7)}
\][/tex]
So, the range of [tex]\( g(x) \)[/tex] is:
[tex]\[
[-7, -6]
\][/tex]
### Final Result:
Domain: [tex]\([-9, 11]\)[/tex]
Range: [tex]\([-7, -6]\)[/tex]
Therefore, the final answer in interval notation is:
[tex]\[
\text{Domain: } [-9, 11] \quad \text{Range: } [-7, -6]
\][/tex]
