Question

Find the inverse of y = x^3 +2x^2
Please, help

Answer 1

\(y\) is not one-to-one

Answer 2

but how to prove? since it it from precalculus course, I don't know how to explain.

Answer 3

\[x^3+2x^2=x^2(x+2)\]
has two roots, \(x=0\) (multiplicity 2) and \(x=-2\) (multiplicity 1). Graphically, the curve crosses the x-axis once at \(x=-2\), touches the x-axis again at \(x=0\), but remains positive for all \(x>-2\). This implies there are repeated output values between \(-2\) and \(0\).

Answer 4

For \(x_1,x_2\in[-2,0]\), with \(x_1\neq x_2\), you have some that satisfy \(f(x_1)=f(x_2)\), so \(y=f(x)\) cannot be injective.