Question

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

Answer 1

`f^(-1)(x)=sqrt(x-3)`

Using the Vertical Line Test, `x^2+3` and its inverse are functions

Explanation:

We know `x^2+3` is a quadratic, which graphs as a parabola, which we know passes the Vertical Line Test. This is what makes it a function.

We essentially have

`y=x^2+3`

And the first step in finding the inverse of this function is switching `y` and `x`. We now have

`x=y^2+3`

Now, we solve for `y`. Let's subtract `3` from both sides to get

`x-3=y^2`

Taking the square root of both sides, we get

`y=sqrt(x-3)`

Since we solved for `y`, we have found our inverse. This is also equal to

`f^(-1)(x)=sqrt(x-3)`

Where `f^(-1)(x)` means "inverse".

The base function of our inverse is `sqrtx`, which we also know it is a function from its graph (passes VLT).

Hope this helps!