Question

Question #35453

Answer 1

Because, for any given y value, there are 2 possible x values, a quadratic equation does not have an inverse; more about this in the explanation.

Explanation:

To formally attempt an inverse, we substitute `f^-1(x)` for every x into the given function:

`f(f^-1(x)) = (f^-1(x))^2-3/2`

By definition, the left side becomes x:

`x = (f^-1(x))^2-3/2`

Add `3/2` to both sides:

`x+ 3/2= (f^-1(x))^2`

When we take the square root of both sides, we must split into 2 equations:

`f^-1(x)= sqrt(x+ 3/2) and f^-1(x)= -sqrt(x+ 3/2)`

This means that (except for `x = 3/2`) every value of x maps to two values. An inverse must map a single y value to a single x value in the same way that a function must map a single x value to a single y value. Therefore, no inverse.