To find the inverse function, substitute x for y in the equation y = f(x), and vice versa. Then, solve the resultant equation for y.
The given function is `y = f(x) = (2x+5)/(x+4)` .
Substituting x for y and y for x results in
`x = (2y+5)/(y+4)` .
To solve this equation for y, first multiply both sides by the denominator, y + 4:
x(y+4) = 2y + 5
Open the parenthesis on the left side:
xy + 4x = 2y + 5
Now combine all terms containing y on the left side and the rest of the terms on the right side. To do that, subtract 2y and 4x from both sides:
xy - 2y = 5 - 4x
Factor out y:
y(x - 2) = 5 - 4x
Finally, to isolate y, divide both sides by x - 2:
`y = (5 - 4x)/(x - 2)`
The result is the inverse function of `f(x) = (2x+5)/(x+4)` , also denoted as
`f^(-1)(x) = (5-4x)/(x-2)` .