Reflecting a function over the [tex]\( y \)[/tex]-axis involves replacing every [tex]\( x \)[/tex] in the function with [tex]\(-x\)[/tex].
The original function is:
[tex]\[ f(x) = 5 \left(\frac{1}{5}\right)^x \][/tex]
To reflect this function over the [tex]\( y \)[/tex]-axis, we replace [tex]\( x \)[/tex] with [tex]\(-x\)[/tex]:
[tex]\[ f(-x) = 5 \left(\frac{1}{5}\right)^{-x} \][/tex]
Next, we simplify the expression [tex]\( \left(\frac{1}{5}\right)^{-x} \)[/tex]:
[tex]\[ \left(\frac{1}{5}\right)^{-x} = \left(\frac{5}{1}\right)^x = 5^x \][/tex]
So the reflected function becomes:
[tex]\[ f(-x) = 5 \cdot 5^x = 5(5^x) \][/tex]
Thus, the correct equation that represents the reflected function is:
[tex]\[ f(x) = 5(5)^x \][/tex]
Let's verify the options given:
1. [tex]\( f(x) = \frac{1}{5}(5)^{-x} \)[/tex] – This is not correct because it does not match the form we derived.
2. [tex]\( f(x) = \frac{1}{5} \frac{1}{5}\left(\frac{1}{5}\right)^x \)[/tex] – This is not correct because its form does not match the reflected function.
3. [tex]\( f(x) = 5 \left(\frac{1}{5}\right)^{-x} \)[/tex] – This can be simplified to [tex]\( 5(5^x) \)[/tex], which is correct.
4. [tex]\( f(x) = 5(5)^x \)[/tex] – This matches our simplified result, so it is correct.
5. [tex]\( f(x) = 5(5)^{-x} \)[/tex] – This is not correct because it involves a negative exponent, implying a completely different transformation.
Therefore, the equations that correctly represent the reflected function are:
[tex]\[ f(x) = 5 \left(\frac{1}{5}\right)^{-x} \text{ and } f(x) = 5(5)^x \][/tex]
