Certainly! Let's solve this problem step-by-step:
Given the original function:
[tex]\[ f(x) = \sqrt{x} \][/tex]
We need to perform two transformations:
1. Reflect the function in the [tex]\( y \)[/tex]-axis.
2. Apply a horizontal stretch by a factor of [tex]\(\frac{3}{4}\)[/tex].
### Step 1: Reflecting in the [tex]\( y \)[/tex]-axis
Reflecting in the [tex]\( y \)[/tex]-axis means that every [tex]\( x \)[/tex] value in the function is replaced with [tex]\(-x \)[/tex]. Therefore, the function becomes:
[tex]\[ g(x) = f(-x) = \sqrt{-x} \][/tex]
### Step 2: Horizontal Stretch by a factor of [tex]\(\frac{3}{4}\)[/tex]
A horizontal stretch by a factor of [tex]\(\frac{3}{4}\)[/tex] means we need to compress the [tex]\( x \)[/tex] values. To apply this, we replace [tex]\( x \)[/tex] with [tex]\(\frac{4}{3}x \)[/tex]. Thus, the function becomes:
[tex]\[ j(x) = \sqrt{\frac{4}{3}(-x)} \][/tex]
### Simplifying the Function
To simplify the function, we can factor out the constant outside the square root:
[tex]\[ j(x) = \sqrt{\frac{4}{3}(-x)} = \sqrt{\frac{4}{3}} \cdot \sqrt{-x} \][/tex]
### Numerical Approximation of [tex]\(\sqrt{\frac{4}{3}}\)[/tex]
The numerical value of [tex]\(\sqrt{\frac{4}{3}}\)[/tex] is approximately [tex]\(1.1547\)[/tex]. Therefore, we can rewrite the function as:
[tex]\[ j(x) = 1.1547 \cdot \sqrt{-x} \][/tex]
Hence, the new function after applying both the reflection and the horizontal stretch is:
[tex]\[ j(x) = 1.1547 \sqrt{-x} \][/tex]
