Sure, let's solve the problem of finding the domain and range of the function [tex]\( g(x) = 2f(x+5) + 3 \)[/tex].
### Step 1: Understanding the Transformations
1. Horizontal Shift: The function [tex]\( g(x) = 2f(x+5) + 3 \)[/tex] indicates a horizontal shift to the left by 5 units, which affects the domain of [tex]\( f(x) \)[/tex].
2. Vertical Stretch and Shift: The function [tex]\( g(x) = 2f(x+5) + 3 \)[/tex] also involves a vertical stretching by a factor of 2 and an upward shift by 3 units, which affects the range of [tex]\( f(x) \)[/tex].
### Step 2: Finding the Domain of [tex]\( g(x) \)[/tex]
Since the input [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex] is shifted by 5 units to the left ([tex]\( x \to x + 5 \)[/tex]), the domain of [tex]\( g(x) \)[/tex] will be obtained by subtracting 5 from each endpoint of the domain of [tex]\( f(x) \)[/tex].
The original domain of [tex]\( f(x) \)[/tex] is [tex]\([-9, 2]\)[/tex].
To find the new domain for [tex]\( g(x) \)[/tex]:
[tex]\[
\text{New domain} = [-9 - 5, 2 - 5] = [-14, -3]
\][/tex]
### Step 3: Finding the Range of [tex]\( g(x) \)[/tex]
The range of [tex]\( g(x) \)[/tex] involves both a vertical stretch and a vertical shift from the range of [tex]\( f(x) \)[/tex]. Specifically, [tex]\( g(x) \)[/tex] is stretched by a factor of 2 and then shifted up by 3 units.
The original range of [tex]\( f(x) \)[/tex] is [tex]\([2, 10]\)[/tex].
To find the new range for [tex]\( g(x) \)[/tex]:
1. Stretch the range by a factor of 2: Multiply each value in the original range by 2.
[tex]\[
\text{Stretched range} = [2 \times 2, 2 \times 10] = [4, 20]
\][/tex]
2. Shift up by 3 units: Add 3 to each value in the stretched range.
[tex]\[
\text{New range} = [4 + 3, 20 + 3] = [7, 23]
\][/tex]
### Final Answer
Putting it all together, we have the following:
- Domain: [tex]\([-14, -3]\)[/tex]
- Range: [tex]\([7, 23]\)[/tex]
So, the domain and range of [tex]\( g(x) = 2f(x+5) + 3 \)[/tex] are:
Domain: [tex]\([-14, -3]\)[/tex]
Range: [tex]\([7, 23]\)[/tex]
