To find the inverse function of [tex]\( f(x) = 5x \)[/tex], denoted as [tex]\( f^{-1}(x) \)[/tex], follow these steps:
1. Express the Function:
Start with the function given:
[tex]\[
f(x) = 5x
\][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
For clarity, let's rewrite the function as:
[tex]\[
y = 5x
\][/tex]
3. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
We need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. To do this, we solve the equation for [tex]\( x \)[/tex]:
[tex]\[
y = 5x
\][/tex]
Divide both sides by 5 to isolate [tex]\( x \)[/tex]:
[tex]\[
x = \frac{y}{5}
\][/tex]
4. Write the Inverse Function:
Since [tex]\( x \)[/tex] is expressed in terms of [tex]\( y \)[/tex], we can now write the inverse function by replacing [tex]\( y \)[/tex] with [tex]\( x \)[/tex]:
[tex]\[
f^{-1}(x) = \frac{x}{5}
\][/tex]
This can also be written as:
[tex]\[
f^{-1}(x) = \frac{1}{5}x
\][/tex]
So, the inverse function of [tex]\( f(x) = 5x \)[/tex] is [tex]\( f^{-1}(x) = \frac{1}{5}x \)[/tex].
Therefore, the correct answer is:
[tex]\[
f^{-1}(x) = \frac{1}{5} x
\][/tex]
