Answer :
To determine the inverse function [tex]\( f^{-1}(x) \)[/tex] for the given function [tex]\( f(x) = 5x \)[/tex]:
1. Define the Function: We start with the given function:
[tex]\[ f(x) = 5x \][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]: For finding the inverse, we replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 5x \][/tex]
3. Solve for [tex]\( x \)[/tex] in Terms of [tex]\( y \)[/tex]: To find the inverse function, we need to solve this equation for [tex]\( x \)[/tex]:
[tex]\[ y = 5x \][/tex]
[tex]\[ x = \frac{y}{5} \][/tex]
4. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] in the Inverse Function: We express the result of the inverse function with [tex]\( x \)[/tex] instead of [tex]\( y \)[/tex]. Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{x}{5} \][/tex]
5. Simplify the Expression: To express it in a simplified form:
[tex]\[ f^{-1}(x) = \frac{1}{5}x \][/tex]
6. Identify the Correct Option: Comparing this result with the given options, the correct option is:
[tex]\[ f^{-1}(x) = \frac{1}{5}x \][/tex]
Therefore, [tex]\( f^{-1}(x) = \frac{1}{5}x \)[/tex]. Thus, the correct answer is:
[tex]\[ \boxed{3 \text{ (Corresponding to } f^{-1}(x) = \frac{1}{5}x \text{ )}} \][/tex]
1. Define the Function: We start with the given function:
[tex]\[ f(x) = 5x \][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]: For finding the inverse, we replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 5x \][/tex]
3. Solve for [tex]\( x \)[/tex] in Terms of [tex]\( y \)[/tex]: To find the inverse function, we need to solve this equation for [tex]\( x \)[/tex]:
[tex]\[ y = 5x \][/tex]
[tex]\[ x = \frac{y}{5} \][/tex]
4. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] in the Inverse Function: We express the result of the inverse function with [tex]\( x \)[/tex] instead of [tex]\( y \)[/tex]. Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{x}{5} \][/tex]
5. Simplify the Expression: To express it in a simplified form:
[tex]\[ f^{-1}(x) = \frac{1}{5}x \][/tex]
6. Identify the Correct Option: Comparing this result with the given options, the correct option is:
[tex]\[ f^{-1}(x) = \frac{1}{5}x \][/tex]
Therefore, [tex]\( f^{-1}(x) = \frac{1}{5}x \)[/tex]. Thus, the correct answer is:
[tex]\[ \boxed{3 \text{ (Corresponding to } f^{-1}(x) = \frac{1}{5}x \text{ )}} \][/tex]
