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

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Answer 1 · 2018-05-30T20:33:55

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

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

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!