To determine the function that represents the reflection of [tex]\( f(x) = 5 \left( \frac{1}{5} \right)^x \)[/tex] over the [tex]\( y \)[/tex]-axis, we need to understand the transformation involved in this reflection. Reflecting a function over the [tex]\( y \)[/tex]-axis is accomplished by replacing [tex]\( x \)[/tex] with [tex]\( -x \)[/tex] in the function.
Let's start with the given function:
[tex]\[ f(x) = 5 \left( \frac{1}{5} \right)^x \][/tex]
Next, we reflect this function over the [tex]\( y \)[/tex]-axis by substituting [tex]\( -x \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[ f(-x) = 5 \left( \frac{1}{5} \right)^{-x} \][/tex]
Now let's simplify [tex]\( 5 \left( \frac{1}{5} \right)^{-x} \)[/tex]:
Recall that [tex]\( \left( \frac{1}{5} \right)^{-x} \)[/tex] is equivalent to:
[tex]\[ \left( \frac{1}{5} \right)^{-x} = \left( \frac{1}{5} \right)^{-1 \cdot -x} = \left( 5 \right)^{x} \][/tex]
Therefore, the reflected function simplifies to:
[tex]\[ f(-x) = 5 \left( 5 \right)^{x} = 5 \left( 5 \right)^{-x} \][/tex]
We need to find which of the given options match this reflected form:
1. [tex]\( f(x) = \frac{1}{5}(5)^{-x} \)[/tex]
2. [tex]\( f(x) = \frac{1}{5} \frac{1}{5}\left(\frac{1}{5}\right)^x \)[/tex]
3. [tex]\( f(x) = 5\left(\frac{1}{5}\right)^{-x} \)[/tex]
4. [tex]\( f(x) = 5(5)^x \)[/tex]
5. [tex]\( f(x)=5(5)^{-x} \)[/tex]
Let's analyze each option one by one.
1. [tex]\( \frac{1}{5}(5)^{-x} \)[/tex]
- Simplifies to: [tex]\( \frac{1}{5} \times \frac{1}{(5^x)} = \frac{1}{5^{x+1}} \)[/tex]
- This does not match our reflected function.
2. [tex]\( \frac{1}{5} \frac{1}{5}\left(\frac{1}{5}\right)^x \)[/tex]
- Simplifies to: [tex]\( \left(\frac{1}{5}\right)^2 \times (5^x) = \frac{1}{25} \times 5^x \)[/tex]
- This does not match our reflected function.
3. [tex]\( 5\left(\frac{1}{5}\right)^{-x} \)[/tex]
- Simplifies to: [tex]\( 5 \times 5^x = 5^{x+1} \)[/tex]
- This does match our reflected function [tex]\( 5(5)^x \)[/tex].
4. [tex]\( 5(5)^x \)[/tex]
- This matches our reflected function [tex]\( 5(5)^x \)[/tex].
5. [tex]\( 5(5)^{-x} \)[/tex]
- This also matches our reflected function [tex]\( 5(5)^{-x} \)[/tex].
So, the two equations that represent the reflected function most appropriately are:
[tex]\[ f(x) = 5 \left( \frac{1}{5} \right)^{-x} \][/tex]
[tex]\[ f(x) = 5(5)^{-x} \][/tex]
