Inverse function of y = x(x+1)?

Question

Can someone please help me with this problem and also explain how you got the answer? The domain is x>(or equal to) -1/2 Thanks!

Answer 1

Rewriting your equation as a parabola

y=x^2+x

y=x^2+x+1/4-1/4

y=(x+1/2)^2 - 1/4

The domain is all reals, and the range is y≥-1/4

So the inverse, should it exist, would have opposite domain and range:

The domain is x≥-1/4 and the range are all reals.

Replacing x's with y's, and y's with x's

x = (y+1/2)^2-1/4

Then solve for y

x+1/4=(y+1/2)^2

y+1/2=±√(x+1/4)

y=-1/2±√(x+1/4)

but this fails the vertical line test, so it is not a function, so the inverse does not exist, which means the domain and range are meaningless.