# How do you find the domain and range of # f(x)=x^2+x#?

##### 1 Answer

Domain:

Range:

#### Explanation:

Given:

#f(x) = x^2+x#

As with any polynomial, this is well defined for all values of

One way of finding the domain is to complete the square:

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

Note that:

#(x+1/2)^2 >= 0#

for any Real value of

So the minimum value of

#f(-1/2) = 0^2-1/4 = -1/4#

Since

One way of proving that goes as follows.

Let:

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

Add

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

Transpose and take the square root of both sides, allowing for both positive and negative square roots to get:

#x+1/2 = +-sqrt(y+1/4)#

Subtract

#x = -1/2+-sqrt(y+1/4)#

So, provided