Sure, let's derive the inverse of the function [tex]\( f(x) = 5x \)[/tex] step by step.
### Step-by-Step Solution:
1. Start with the given function:
[tex]\[
f(x) = 5x
\][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex] for easier manipulation:
[tex]\[
y = 5x
\][/tex]
3. To find the inverse function [tex]\( f^{-1}(x) \)[/tex], solve this equation for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
y = 5x
\][/tex]
4. Isolate [tex]\( x \)[/tex] by dividing both sides of the equation by 5:
[tex]\[
x = \frac{y}{5}
\][/tex]
5. Now, express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex], which can now be replaced back with [tex]\( x \)[/tex] since we're defining the inverse function:
[tex]\[
f^{-1}(x) = \frac{x}{5}
\][/tex]
6. Simplify the expression:
[tex]\[
f^{-1}(x) = \frac{1}{5}x
\][/tex]
Hence, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{1}{5}x
\][/tex]
The correct answer is:
[tex]\[
f^{-1}(x) = \frac{1}{5} x
\][/tex]
