Question

How do you find the inverse of `y=4-x^2+3x` and is it a function?

Answer 1

`x=(3+-sqrt(25-y))/2`
which is not a function since it results in multiple solutions.

Explanation:

Given:
`color(white)("XXX")y=4-x^2+3x`

Re-arrange the equation to solve for `x` in terms of `y`

`color(white)("XXX")-y=x^2-3x-4`

`color(white)("XXX")-y= (x^2-3xcolor(red)(+(3/2)^2))-4color(red)(-9/4)`

`color(white)("XXX")-y=(x-3/2)^2-25/4`

`color(white)("XXX")(x-3/2)^2=25/4-y`

`color(white)("XXX")x-3/2=+-sqrt(25-4y)/2`

`color(white)("XXX")x=(3+-sqrt(25-4y))/2`

In order to maintain `y` as the dependent variable (and `x` as the independent variable) it is common to exchange the `x` and `y` variables at this point.
In this case the inverse would be written as:
`color(white)("XXX")y=(3+-sqrt(25-x))/2`

Sometimes the `y` is written in some modified form such as `y'` to avoid confusing it with the `y` in the original equation.

Check with your instructor for the form she/he wants you to use.