To find the reflected function of \( f(x) = 5\left(\frac{1}{5}\right)^x \) over the \( y \)-axis, we need to replace \( x \) with \( -x \) in the original function.
1. Original Function:
\( f(x) = 5\left(\frac{1}{5}\right)^x \)
2. Reflect over the \( y \)-axis:
Substitute \( -x \) for \( x \):
\( f(-x) = 5\left(\frac{1}{5}\right)^{-x} \)
3. Simplify \( \left(\frac{1}{5}\right)^{-x} \):
Recall that \( \left(\frac{1}{5}\right)^{-x} \) is equivalent to \( 5^x \)
Therefore, \( f(-x) = 5(5^x) \)
4. Identify the Equations:
Now, the simplified reflected function is:
\( f(-x) = 5(5)^x \)
The function after reflection can be written in two forms:
- The direct substitution form before simplification:
\( f(-x) = 5\left(\frac{1}{5}\right)^{-x} \)
- The simplified form:
\( f(-x) = 5(5)^x \)
So, the correct options that represent the reflected function are:
- \( f(x) = 5\left(\frac{1}{5}\right)^{-x} \)
- \( f(x) = 5(5)^x \)
These match the following options:
- \( f(x)=5\left(\frac{1}{5}\right)^{-x} \)
- \( f(x)=5(5)^x \)
Therefore, the correct answers are:
[tex]\[ f(x)=5\left(\frac{1}{5}\right)^{-x} \][/tex]
[tex]\[ f(x)=5(5)^x \][/tex]
