Question

Please help!!

Determine whether the inverse of F(x) is a function.

Answer 1

f⁻¹(x) is not a function

Explanation:

A function can only have one output value per input value. Multiple input values can have the same output value (horizontally), but multiple output values cannot have the same input value (vertically).

For example:

`f(x)=x^2\\f(-2)=-2^2=4\\f(2)=2^2=4`

This function has 2 inputs with the same output. That is valid.

However, if you take the inverse of that function:

`f^(-1)(x)=√(x)\\f^(-1)(2)=√(2)=1.414\\f^(-1)(-2)=√(-2)=√(2)i`

f⁻¹(-2) is an imaginary number. It doesn't exist, and so the inverse of f(x) is still a function because there is no vertical overlap. This is shown more clearly in the attached image.

The graph of the inverse of a function is just mirrored over y = x. This is also shown in the attached image. Any horizontal overlap the function may have becomes vertical overlap, and if those values still exist, the function cannot be a function.

In the function in your question, f(x) has some horizontal overlap around x = 0 and x = 8. If this was mirrored over y = x, the function at x = 8 would have 2 outputs, y = 0 and y = 8. As a result, it cannot be a function.

Let me know if I need to explain any part of that more clearly.