What is the domain and range of #y = (x^2 + 4x + 4)/( x^2 - x - 6)#?

Question

Genesis Allen · Accepted Answer

See below.

Let's factor the numerator and denominator of the function to see if we can simplify it before taking any further action. #((x+2)(x+2))/((x+2)(x-3))# You can see that one of the #x+2# terms cancel: #(x+2)/(x-3)# The domain of a function is all of the #x#values (horizontal axis) that will give you a valid y-value (vertical axis) output. Since the function given is a fraction, dividing by #0# will not yield a valid #y# value. To find the domain, let's set the denominator equal to zero and solve for #x#. The value(s) found will be excluded from the range of the function. #x-3=0# #x=3# So, the domain is all real numbers EXCEPT #3#. In set notation, the domain would be written as follows: #(-oo,3)uu(3,oo)# The range of a function is all of the #y#-values that it can take on. Let's graph the function and see what the range is. graph{ (10, 10, -5, 5]} graph{(x+2)/(x-3) We can see that as #x# approaches #3#, #y# approaches #oo#. We can also see that as #x# approaches #oo#, #y# approaches #1#. The range would be expressed as follows in set notation: #(-oo,1)uu(1,oo)#

Isaiah Anthony · Answer

undefined Domain: All real numbers except x = 3 and x = -2.
Range: All real numbers.