What is the domain and range of (x^2+2)/(x+4)x2+2x+4?

1 Answer
Apr 6, 2018

The domain is x in RR-{-4}. The range is y in (-oo, -16.485] uu [0.485, +oo)

Explanation:

The denominator is !=0

x+4!=0

x!=-4

The domain is x in RR-{-4}

To find the range, proceed as follws

Let y=(x^2+2)/(x+4)

y(x+4)=x^2+2

x^2-yx+2-4y=0

This is a quadratic equation in x^2 and in order to have solutions

the discriminant Delta>=0

Therefore

Delta=(-y)^2-4(1)(2-4y)>=0

y^2-16y-8>=0

The solutions are

y=(-16+-sqrt((-16)^2-4(1)(-8)))/2=(-16+-16.97)/2

y_1=-16.485

y_2=0.485

The range is y in (-oo, -16.485] uu [0.485, +oo)

graph{(x^2+2)/(x+4) [-63.34, 53.7, -30.65, 27.85]}