How do you find the inverse of f(x) = (2x-1)/(x-1) and is it a function?

1 Answer
May 14, 2016
  • f^-1(x)=(-x+1)/(2-x)
  • yes, it is a function

Explanation:

Determining the Inverse Function
Given,

f(x)=(2x-1)/(x-1)

Substitute y for f(x).

y=(2x-1)/(x-1)

Swap the x and y.

x=(2y-1)/(y-1)

Solve for y.

x(y-1)=2y-1

xy-x=2y-1

2y-xy=-x+1

Factor out y from the left side.

y(2-x)=-x+1

y=(-x+1)/(2-x)

Rewrite y as f^-1(x).

color(green)(|bar(ul(color(white)(a/a)color(black)(f^-1(x)=(-x+1)/(2-x))color(white)(a/a)|)))

Determining Whether the Inverse Function Is a Function
Graphically, f^-1(x)=(-x+1)/(2-x) would look like:

graph{(-x+1)/(2-x) [-10, 10, -5, 5]}

In the graph above, you can see that the x and y values approach the vertical and horizontal asymptotes. Since it resembles that of an exponential graph, there is only one y value for an x value.

:., the inverse function is a function.