Question

For which of the following functions does `f(x) = f^-1(x)`: A) `f(x) = (2x-1)/2`, B) `f(x) = 10-x`, C) `f(x) = x-4`, D) `f(x) = x+4`?

Answer 1

See below.

Explanation:

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

Now by inspection

`10-(10-x) = x = f(f(x))`

Answer 2

Answer B: `f^-1(x) = 10-x`

Explanation:

To find the inverse of a given function, simply swap `x` for `f^-1(x)` and `f(x)` for `x` and solve for `f^-1(x)`

Let's try each of the options in turn.

A.
`f(x) = (2x-1)/2`

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

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

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

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

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

B.
`f(x) = 10-x`

`x = 10-f^-1(x)`

`f^-1(x) = 10-x`

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

C.
`f(x) = x-4`

`x= f^-1(x) -4`

`f^-1(x) = x+4`

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

D.
`f(x) = x+4`

`x= f^-1(x) +4`

`f^-1(x) = x-4`

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

Hence, answer B is the only one that satisfies `f^-1(x) = f(x)`