To find the inverse function [tex]\( f^{-1}(x) \)[/tex] for the function [tex]\( f(x) = \frac{x-7}{5} \)[/tex], follow these detailed steps:
1. Start with the original function:
[tex]\[
y = f(x) = \frac{x-7}{5}
\][/tex]
2. Interchange the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = \frac{y-7}{5}
\][/tex]
3. Solve the new equation for [tex]\( y \)[/tex]:
- Multiply both sides by 5 to get rid of the denominator:
[tex]\[
5x = y - 7
\][/tex]
- Add 7 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = 5x + 7
\][/tex]
4. Thus, the inverse function is:
[tex]\[
f^{-1}(x) = 5x + 7
\][/tex]
So the inverse function [tex]\( f^{-1}(x) \)[/tex] is [tex]\( 5x + 7 \)[/tex].
