Question

How can i find the inverse for this function? help me please

Answer 1

Substitute `f^-1(x)` for x.
Use the property `x = f(f^-1(x))`
Solve for `f^-1(x)`
Verify the inverse by asserting `x = f(f^-1(x))` and `x = f^-1(f(x))`

Explanation:

Given: `f(x) = 1/(x-2)`

Substitute `f^-1(x)` for x.

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

Use the property `x = f(f^-1(x))`

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

Solve for `f^-1(x)`.

Multiply both sides by `(f^-1(x)-2)/x`

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

Add 2 to both sides:

`f^-1(x) = 1/x + 2`

Verify the inverse by asserting `x = f(f^-1(x))` and `x = f^-1(f(x))`

Check `f^-1(f(x))`:

`f^-1(f(x)) = 1/(f(x)) + 2`

`f^-1(f(x)) = 1/(1/(x-2)) + 2`

`f^-1(f(x)) = x-2 + 2`

`f^-1(f(x)) = x`

Check `f(f^-1(x))`:

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

`f(f^-1(x)) = 1/(1/x + 2-2)`

`f(f^-1(x)) = 1/(1/x)`

`f(f^-1(x)) = x`

Both are verified, therefore, `f^-1(x) = 1/x + 2`