Let's tackle the problem step by step with the given transformations.
### Step 1: Original Function
The original function given is:
[tex]\[ f(x) = \sqrt{x - 2} + 1 \][/tex]
### Step 2: Vertical Compression by a Factor of 3
A vertical compression by a factor of [tex]\( \frac{1}{3} \)[/tex] means we multiply the entire function by [tex]\( \frac{1}{3} \)[/tex]:
[tex]\[ g(x) = \frac{1}{3} f(x) = \frac{1}{3} (\sqrt{x - 2} + 1) \][/tex]
### Step 3: Horizontal Translation Left by 4
A horizontal translation left by 4 units means we replace [tex]\( x \)[/tex] with [tex]\( x + 4 \)[/tex] in the function [tex]\( g(x) \)[/tex]:
[tex]\[ h(x) = g(x + 4) = \frac{1}{3} (\sqrt{(x + 4) - 2} + 1) = \frac{1}{3} (\sqrt{x + 2} + 1) \][/tex]
### Step 4: Vertical Translation Up by 1
A vertical translation up by 1 unit means we add 1 to the entire function [tex]\( h(x) \)[/tex]:
[tex]\[ k(x) = h(x) + 1 = \frac{1}{3} (\sqrt{x + 2} + 1) + 1 \][/tex]
### Conclusion
Thus, the function after all three transformations is:
[tex]\[ k(x) = \frac{1}{3} (\sqrt{x + 2} + 1) + 1 \][/tex]
### Example Evaluation
To evaluate this transformed function at a specific value, let's take [tex]\( x = 6 \)[/tex]:
1. Substitute [tex]\( x = 6 \)[/tex] into the transformed function:
[tex]\[ k(6) = \frac{1}{3} (\sqrt{6 + 2} + 1) + 1 \][/tex]
2. Calculate inside the square root:
[tex]\[ \sqrt{6 + 2} = \sqrt{8} \][/tex]
3. Add 1:
[tex]\[ \sqrt{8} + 1 \][/tex]
4. Apply the vertical compression factor [tex]\( \frac{1}{3} \)[/tex]:
[tex]\[ \frac{1}{3} (\sqrt{8} + 1) \][/tex]
5. Add 1 for the vertical translation:
[tex]\[ \frac{1}{3} (\sqrt{8} + 1) + 1 \][/tex]
After simplifying these steps, we get the numerical result:
[tex]\[ 2.2761423749153966 \][/tex]
Therefore, the value of the transformed function at [tex]\( x = 6 \)[/tex] is approximately [tex]\( 2.2761423749153966 \)[/tex].
