Question

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

Answer 1

• `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.