Let's begin by addressing the transformation applied to the function [tex]\(f(x)\)[/tex] to obtain [tex]\(g(x)\)[/tex]. The original domain of [tex]\(f(x)\)[/tex] is [tex]\([-1, 4]\)[/tex] and the range is [tex]\([6, 7]\)[/tex].
First, let's determine the domain of the transformed function [tex]\(g(x) = -f\left(\frac{1}{4}(x + 5)\right)\)[/tex].
1. Identify the input [tex]\(u\)[/tex] to the function [tex]\(f\)[/tex]:
[tex]\[
u = \frac{1}{4}(x + 5)
\][/tex]
2. Since [tex]\(f(u)\)[/tex] has a domain of [tex]\([-1, 4]\)[/tex], the transformed variable [tex]\(\frac{1}{4}(x + 5)\)[/tex] must also be within this interval:
[tex]\[
-1 \leq \frac{1}{4}(x + 5) \leq 4
\][/tex]
3. Solve for [tex]\(x\)[/tex] in the inequalities separately:
For the lower bound:
[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]\[
-9 \leq x
\][/tex]
For the upper bound:
[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]
Combining these results, the domain of [tex]\(g(x)\)[/tex] is:
[tex]\[
-9 \leq x \leq 11
\][/tex]
In interval notation, this is:
[tex]\[
\text{Domain: } [-9, 11]
\][/tex]
Next, let's determine the range of the function [tex]\(g(x)=-f\left(\frac{1}{4}(x+5)\right)\)[/tex].
1. The range of [tex]\(f(u)\)[/tex] is [tex]\([6, 7]\)[/tex].
2. The function [tex]\(g(x)\)[/tex] negates the values of [tex]\(f(u)\)[/tex]. Therefore, if [tex]\(f(u)\)[/tex] ranges from 6 to 7, the negated values will range from [tex]\(-6\)[/tex] to [tex]\(-7\)[/tex].
3. Since [tex]\(-7\)[/tex] is smaller than [tex]\(-6\)[/tex], we write the range in the correct interval notation as:
[tex]\[
\text{Range: } [-7, -6]
\][/tex]
So, the domain and range of the function [tex]\(g(x)=-f\left(\frac{1}{4}(x+5)\right)\)[/tex] are:
[tex]\[
\text{Domain: } [-9, 11], \text{ Range: } [-7, -6]
\][/tex]
