To find the inverse of the function [tex]\( f(x) = \frac{x+4}{x} \)[/tex], we need to follow these steps:
1. Express the given function in terms of [tex]\( y \)[/tex]:
Begin by writing the function as an equation where [tex]\( y = f(x) \)[/tex]:
[tex]\[
y = \frac{x+4}{x}
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
Rearrange the equation to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
y = \frac{x+4}{x}
\][/tex]
Multiply both sides by [tex]\( x \)[/tex] to eliminate the fraction:
[tex]\[
yx = x + 4
\][/tex]
Move all terms involving [tex]\( x \)[/tex] to one side of the equation:
[tex]\[
yx - x = 4
\][/tex]
Factor out [tex]\( x \)[/tex] on the left-hand side:
[tex]\[
x(y-1) = 4
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{4}{y-1}
\][/tex]
3. Rewrite the inverse function:
Since we originally started with [tex]\( y = f(x) \)[/tex], after solving for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex], we now rename [tex]\( y \)[/tex] back to [tex]\( x \)[/tex] to find the inverse function [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
f^{-1}(x) = \frac{4}{x-1}
\][/tex]
Thus, the inverse of the function [tex]\( f(x) = \frac{x+4}{x} \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{4}{x-1}
\][/tex]
This completes our step-by-step solution for finding the inverse of the given function.
