What is the domain and range of the following 
rational expression? 
f(x)=x+3
_____
x^2-5x-66


f(x) = 

Factor the denominator:

f(x) = 

The domain contains all numbers which 
can be substituted for x, and an answer
is defined.  Denominators cannot be 0
so x cannot be any number that will
cause the denominator to be 0.

DENOMINATOR = (x-11)(x+2)

So x cannot equal to 11 or -2

So the domain is 

 

or in interval notation (-,-2)U(-2,11)U(11,)

Now we have to find the range.

To do that we first replace f(x) by y and solve for x:

f(x) = 

y = 

Multiply both sides by the LCD = 

 = 

 = 

Get 0 on the right:

 = 

Get the x terms together:

 = 

Factor x out of the second and third terms

 = 

factor -3 out of the last two terms on the left:

 = 

This is a quadratic equation in x with

a = y, b = -(5y+1), c = -3(22y+1)

We could solve for x, but we only
need the discriminant b²-4ac to be
non-negative, so

b²-4ac = [-(5y+1)]² - 4(y)[-3(22y+1)] =

25y² + 10y + 1 + 12y(22y+1) =

25y² + 10y + 1 + 264y² + 12y =

289y² + 22y + 1 > = 0

The left side has discriminant 22²-4(289)(1) = -672

thus it has no real zeros and represents a parabola which
is always positive.

Therefore the range is "all real numbers" or (-,)

Edwin