How do you find the domain and range of f(x)=x2+2x+2?

1 Answer
Nov 13, 2017

Domain: xR
Range: f(x)[1,+),R

Explanation:

f(x)=x2+2x+2 is defined for all Real values of x
Domain is R

f(x)=x2+2x+2 has a minimum value of f(x)=1 but no maximum value
Range is [1,+),R

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

How do we know that f(x) has a minimum value of 1?

Since f(x) is a quadratic with a coefficient of x which is greater than zero,
we know that it has a minimum.
We can find this minimum by taking the derivative of f(x), setting that to zero, solving for x, and then using that value to find f(x) at that value:
XXXf'(x)=2x+2=0
XXXx=1
XXXf(x=1)=(1)2+2(1)+2=1

If it helps here is a graph of f(x)=x2+2x+2 to help verify this:
graph{x^2+2x+2 [-3.61, 2.55, -0.512, 2.568]}