To determine which functions represent a reflection over the [tex]\( y \)[/tex]-axis of the given function, we need to understand the effect of reflecting a function over the [tex]\( y \)[/tex]-axis on its formula.
Reflecting a function [tex]\( f(x) \)[/tex] over the [tex]\( y \)[/tex]-axis gives us the new function [tex]\( f(-x) \)[/tex]. Thus, we will evaluate [tex]\( g(-x) \)[/tex] for the given function and compare it with the potential reflection functions.
Given function:
[tex]\[ g(x) = -\frac{1}{2}(4)^x \][/tex]
First, let's determine [tex]\( g(-x) \)[/tex]:
[tex]\[ g(-x) = -\frac{1}{2}(4)^{-x} = -\frac{1}{2} \cdot \frac{1}{4^x} = -\frac{1}{2} \cdot 4^{-x} = -\frac{1}{2} \left(\frac{1}{4}\right)^x \][/tex]
We need to find which of the provided functions match this expression. Let's look at the options one by one:
1. [tex]\( g(x) = 0.5 (4)^{-x} \)[/tex]
- [tex]\( 0.5 \)[/tex] is equivalent to [tex]\( \frac{1}{2} \)[/tex], so rewriting it gives:
[tex]\[ g(x) = \frac{1}{2} (4)^{-x} \][/tex]
- This matches our exploration of [tex]\( g(-x) \)[/tex], but with a positive sign in front.
- Therefore, [tex]\( g(x) = 0.5 (4)^{-x} \)[/tex] is not a reflection of our given function.
2. [tex]\( g(x) = 2 (4)^x \)[/tex]
- This does not have the same structure as either [tex]\( -\frac{1}{2}(4)^x \)[/tex] or [tex]\( -\frac{1}{2}\left(\frac{1}{4}\right)^x \)[/tex].
- Therefore, [tex]\( g(x) = 2 (4)^x \)[/tex] is not a reflection of our given function.
3. [tex]\( g(x) = \frac{1}{2}\left(\frac{1}{4}\right)^x \)[/tex]
- This matches our exploration except for the positive sign.
- Therefore, [tex]\( g(x) = \frac{1}{2}\left(\frac{1}{4}\right)^x \)[/tex] is not a reflection of our given function.
4. [tex]\( g(x) = \frac{1}{2}\left(\frac{1}{4}\right)^{-x} \)[/tex]
- Rewriting [tex]\(\left(\frac{1}{4}\right)^{-x} = (4)^x\)[/tex], so:
[tex]\[ g(x) = \frac{1}{2} (4)^x \][/tex]
- This does not match our expression for [tex]\( g(-x) \)[/tex], as the sign in front does not match.
- Therefore, [tex]\( g(x) = \frac{1}{2}\left(\frac{1}{4}\right)^{-x} \)[/tex] is not a reflection of our given function.
Given the options, there are no functions among the given potential functions that match a perfect reflection over the [tex]\( y \)[/tex]-axis of [tex]\( g(x) = -\frac{1}{2}(4)^x \)[/tex].
Thus, none of the given functions represent a reflection over the [tex]\( y \)[/tex]-axis of the function [tex]\( g(x) = -\frac{1}{2}(4)^x \)[/tex].
