What is the inverse function of f(x)=(x+1)^2?

2 Answers
Sep 22, 2015

The function is not one-to-one. It does not have an inverse function. However,

Explanation:

If we restrict the domain to [-1,oo), then this new function has an inverse:

g(x)=(x+1)^2 with x >= -1

y=(x+1)^2
if and only if
x+1 = sqrty (We do not need +-sqrty because of the restriction on x.)

x=sqrty-1

g^-1(x) = -sqrtx-1

If we restrict the domain to (-oo,-1], we get

h(x) = (x+1)^2 with x <= -1

y=(x+1)^2
if and only if
x+1 = - sqrty (We need only -sqrty because of the restriction on x.)

x= - sqrty-1

h^-1(x) = sqrtx-1

Sep 22, 2015

The function is not one-one and therefore there is not a unique inverse function.

Explanation:

First note that the slope (first derivative) is

f'(x) = 2(x + 1)(1) = 2x + 2 (chain rule).

Also note that

0 < 2 x + 2

requires

-2 < 2x

that is

x > -1

That is, the slope is negative for values of x less than -1 and positive for values of x greater than -1 (and has a stationary point at which the slope is zero at x = -1).

That is, the function is not a strictly rising or a strictly descending one.

That is, it is not bijective (one-one).

That is, there is no single inverse function if the domain is taken as the reals.

The function is strictly descending on the open interval (-oo, -1) and strictly rising on the open interval (-1, oo) so there will be inverse functions on these domains.

Setting y = (x + 1)^2

This implies

(x + 1) = +-sqrt(y)

which in turn implies

x = sqrt(y) - 1

or

x = -sqrt(y) - 1

Considering x = sqrt(y) - 1

Denoting the required inverse function by g(x), this may be rewritten as

g(x) = sqrt(x) - 1

For g(x) to be a real function, the domain must be restricted to values of x equal to or greater than 0.

Considering x = -sqrt(y) - 1

Denoting the required inverse function by h(x), this may be rewritten as

h(x) = - sqrt(x) - 1

For h(x) to be a real function, the domain must be restricted to values of x equal to or greater than 0.

Plotting both of these inverse functions on the same graph will yield a parabola "on its side", with apex at (0, -1). That is, values of x > 0 will correspond to two values (one associated with g(x) and the other associated with h(x), reflecting the fact that f(x) is not a one-one function.