Inverse of $x(x+2)$ given $x\ge -1$

Question

Consider the function: $y=x(x+2)$ . Consider its domain to be $x \geq -1$ .

Graphically it makes sense that the inverse of this function is $-1 + \sqrt{x+1}$.

But how to compute it analytically? Thanks

Answer 1

HINT: Multiply to get $y=x^2+2x$ and then use the quadratic formula to solve for $x$.

Answer 2

Verbally, since the function $f$ defined by $f(x)=x(x+2) = (x+1)^2-1$, we have that $f$

• adds one to its input,
• squares the result,
• and subtracts $1$.

An inverse function will do the opposite things in the opposite order:

• add $1$,
• take a positive or negative square root,
• subtract $1$.

Since we seek to invert a function whose domain includes arbitrarily large positive numbers, we reason that we will take the positive square root. So $$f^{-1}\left(y\right)=\sqrt{y+1}-1\text{.}$$