How to determine if function is one-to-one and to determine its inverse formula. Given : f(x)=(x+4)/(x-5)

Answer 1

first let's find the inverse:
y = (x+4)/(x-5)
to form the inverse , we interchange the x and y variables
x = (y+4)/(y-5)
simplify and solve this new equation for y
xy - 5x = y+4
xy - y = 5x + 4
y(x - 1) = 5x+ 4
y = (5x+4)/(x-1) , so f^-1 (x) = (5x+4)/(x-1)

let's look at their graphs:
https://www.wolframalpha.com/input/?i=plot+y+%3D+%28x%2B4%29%2F%28x-5%29+%2C+y+%3D+%285x%2B4%29%2F%28x-1%29

the 2nd graph has the same scale for the x-axis as the y-axis, and we can clearly see that
one is a reflection of the other in the line y = x
This is a fundamental property of a function and its inverse

Look up "vertical line test" to check if a function is a one-to-one