If [tex]\( F(x) = 8x \)[/tex], which of the following is the inverse of [tex]\( F(x) \)[/tex]?

A. [tex]\( F^{-1}(x) = \frac{8}{x} \)[/tex]

B. [tex]\( F^{-1}(x) = x - 8 \)[/tex]

C. [tex]\( F^{-1}(x) = x + 8 \)[/tex]

D. [tex]\( F^{-1}(x) = \frac{x}{8} \)[/tex]



Answer :

To find the inverse of the function [tex]\( F(x) = 8x \)[/tex], follow these steps:

1. Start by writing the function:
[tex]\[ y = 8x \][/tex]
Here, [tex]\( y \)[/tex] represents the output of the function [tex]\( F(x) \)[/tex].

2. Swap the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = 8y \][/tex]
3. Solve for [tex]\( y \)[/tex]:

To isolate [tex]\( y \)[/tex], divide both sides of the equation by 8:
[tex]\[ y = \frac{x}{8} \][/tex]

Thus, the inverse function [tex]\( F^{-1}(x) \)[/tex] is:
[tex]\[ F^{-1}(x) = \frac{x}{8} \][/tex]

4. Identify the correct option based on the given choices:

- A. [tex]\( F^{-1}(x) = \frac{8}{x} \)[/tex]
- B. [tex]\( F^{-1}(x) = x - 8 \)[/tex]
- C. [tex]\( F^{-1}(x) = x + 8 \)[/tex]
- D. [tex]\( F^{-1}(x) = \frac{x}{8} \)[/tex]

The correct choice is:
[tex]\[ \boxed{D. \ F^{-1}(x) = \frac{x}{8}} \][/tex]