To determine the transformed function from the given modifications applied to the parent function [tex]\( f(x) = 5^x \)[/tex]:
1. Vertical Compression by Factor of 1/2:
The vertical compression of [tex]\( f(x) \)[/tex] by a factor of 1/2 means multiplying [tex]\( f(x) \)[/tex] by 1/2. So, [tex]\( 5^x \)[/tex] becomes [tex]\( \left(\frac{1}{2}\right) \cdot 5^x \)[/tex].
2. Shift to the Left by 3 Units:
Shifting the function to the left by 3 units involves replacing [tex]\( x \)[/tex] with [tex]\( x + 3 \)[/tex]. So, [tex]\( 5^x \)[/tex] transforms to [tex]\( 5^{x + 3} \)[/tex].
3. Shift Up by 2 Units:
Shifting the function up by 2 units involves adding 2 to the entire function. So the function becomes [tex]\( g(x) \)[/tex] = [tex]\( \left(\frac{1}{2}\right)\cdot 5^{x + 3} + 2 \)[/tex].
Now, we compare this form to the given options:
- [tex]\( g(x) = \left(\frac{1}{2}\right) 5^{x-0}+2 \)[/tex]
This function doesn't correctly shift the [tex]\( x \)[/tex] value.
- [tex]\( g(x) = 5^{\left(\frac{\eta}{2}\right) x+3}+2 \)[/tex]
This function incorrectly applies a transformation to the exponent part with [tex]\(\left(\frac{\eta}{2}\right)x\)[/tex].
- [tex]\( g(x) = \left(\frac{1}{2}\right) 5^{x \neq t}+2 \)[/tex]
This function contains an undefined notation [tex]\( x \neq t \)[/tex].
- [tex]\( g(x) = 5^{\left(\frac{1}{2}\right)^{x-1}}+2 \)[/tex]
This function modifies the [tex]\( x \)[/tex] term, applying an incorrect exponent form.
The most accurate transformation of the parent function [tex]\( f(x) = 5^x \)[/tex] given the three specific transformations (vertical compression, left shift, and upward shift) is represented by the function:
[tex]\[ g(x) = \left(\frac{1}{2}\right) \cdot 5^{x + 3} + 2 \][/tex]
Therefore, none of the provided options exactly match the correct transformation. However, if there was a misinterpretation in the options given (e.g., notation errors or typing mistakes), you could verify that the derived function accurately represents the described transformations.
