Question

State whether f^-1 (x) is a function given the equation

Answer 1

We have that
`f(a) = f(b)` if
`a = \pm b`, which does not respect the condition for the existence of an inverse function, which means that
`f^(-1)(x)` does not exist for this function.

Explanation:

Existence of an inverse function:

An inverse function will exist if: f(a) = f(b) only if a = b.

In this question:

`f(a) = -(1)/(7)√(16 - a^2)`

`f(b) = -(1)/(7)√(16 - b^2)`

`-(1)/(7)√(16 - a^2) =-(1)/(7)√(16 - b^2)`

`√(16 - a^2) = √(16 - b^2)`

`(√(16 - a^2))^2 = (√(16 - b^2))^2`

`16 - a^2 = 16 - b^2`

`a^2 = b^2`

`a = \pm b`

We have that
`f(a) = f(b)` if
`a = \pm b`, which does not respect the condition for the existence of an inverse function, which means that
`f^(-1)(x)` does not exist for this function.