the function $f(x)=5(\\frac{1}{5})^x$ is reflected over the $y$-axis. which equations represent the…

the function $f(x)=5(\\frac{1}{5})^x$ is reflected over the $y$-axis. which equations represent the reflected function? select two options.\n$\\square f(x)=\\frac{1}{5}(5)^{-x}$\n$\\square f(x)=\\frac{1}{5}\\frac{1}{5}(\\frac{1}{5})^x$\n$\\square f(x)=5(\\frac{1}{5})^{-x}$\n$\\square f(x)=5(5)^x$\n$\\square f(x)=5(5)^{-x}$

the function $f(x)=5(\\frac{1}{5})^x$ is reflected over the $y$-axis. which equations represent the reflected function? select two options.\n$\\square f(x)=\\frac{1}{5}(5)^{-x}$\n$\\square f(x)=\\frac{1}{5}\\frac{1}{5}(\\frac{1}{5})^x$\n$\\square f(x)=5(\\frac{1}{5})^{-x}$\n$\\square f(x)=5(5)^x$\n$\\square f(x)=5(5)^{-x}$

Answer

Explanation:

Step1: Recall reflection rule

When a function $y = f(x)$ is reflected over the $y$-axis, the transformation is $y=f(-x)$. Given $f(x)=5(\frac{1}{5})^x$, then $f(-x)=5(\frac{1}{5})^{-x}$.

Step2: Simplify the expression

Using the negative - exponent rule $a^{-n}=\frac{1}{a^{n}}$, we have $5(\frac{1}{5})^{-x}=5\times5^{x}=5^{x + 1}$. Also, $5(\frac{1}{5})^{-x}=5\times5^{x}$ and $5(\frac{1}{5})^{-x}$ can be rewritten as $5\times5^{x}=5(5)^{x}$.

Step3: Analyze options

  • Option 1: $f(x)=\frac{1}{5}(5)^{-x}=5^{-1}\times5^{-x}=5^{-x - 1}$, not correct.
  • Option 2: $f(x)=\frac{1}{5}\times\frac{1}{5}(\frac{1}{5})^{x}=\frac{1}{25}(\frac{1}{5})^{x}=(\frac{1}{5})^{x + 2}$, not correct.
  • Option 3: $f(x)=5(\frac{1}{5})^{-x}=5\times5^{x}$, correct.
  • Option 4: $f(x)=5(5)^{x}$, correct.
  • Option 5: $f(x)=5(5)^{-x}=5\times5^{-x}=5^{1 - x}$, not correct.

Answer:

C. $f(x)=5(\frac{1}{5})^{-x}$, D. $f(x)=5(5)^{x}$