To find the inverse of the function [tex]\( F(x) = \frac{2x - 3}{5} \)[/tex], we need to follow these steps:
1. Replace [tex]\( F(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = \frac{2x - 3}{5}
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
- First, multiply both sides of the equation by 5 to get rid of the denominator:
[tex]\[
5y = 2x - 3
\][/tex]
- Next, add 3 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
5y + 3 = 2x
\][/tex]
- Finally, divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{5y + 3}{2}
\][/tex]
3. Rewrite [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] as the function [tex]\( F^{-1}(x) \)[/tex]:
- Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to denote the inverse function:
[tex]\[
F^{-1}(x) = \frac{5x + 3}{2}
\][/tex]
4. Identify the correct option:
- We compare our result with the given options:
A. [tex]\( F^{-1}(x) = \frac{3x + 2}{5} \)[/tex]
B. [tex]\( F^{-1}(x) = \frac{3x + 5}{2} \)[/tex]
C. [tex]\( F^{-1}(x) = \frac{2x + 3}{5} \)[/tex]
D. [tex]\( F^{-1}(x) = \frac{5x + 3}{2} \)[/tex]
- The correct answer matches option D:
[tex]\[
F^{-1}(x) = \frac{5x + 3}{2}
\][/tex]
Therefore, the inverse of [tex]\( F(x) \)[/tex] is option D, [tex]\( F^{-1}(x) = \frac{5x + 3}{2} \)[/tex].
