How do you find the domain and range of #g(x) = x/(x^2 - 16)#?

Question

Eva Adamson · Accepted Answer

The domain of #g(x)# is # x in RR-{-4,4}#.
The range is #g(x) in RR# As you cannot divide by #0#, the denominator is #!=0# Consequently, #x^2-16!=0#, #=># #x!=-4# and #x!=4# The domain of #g(x)# is #x in RR-{-4,4}# To determine the range, follow these steps: Let #y=x/(x^2-16)# #y(x^2-16)=x# #yx^2-x-16y=0# This is a quadratic equation in #x#, and in order to have solutions, the discriminant #>=0# #a=y# #b=-1# #c=-16y# #Delta=b^2-4ac=(-1)^2-4(y)(-16y)=1+64y^2# #AA y in RR#, #=>#, #Delta>=0# Consequently, The range is #g(x) in RR# graph{x/(x^2-16) [-10, 10, 5, 5, 10]}

Julia Adams · Answer

Domain: #(-oo, -4)uu(-4, 4)uu(4, oo)#
Range: #(-oo, oo)# Given: #x/(x^2 - 16)# First factor the denominator since #(x^2-16)# is the difference of squares: #x/(x^2 - 16) = x/((x-4)(x+4))# Find the Domain - valid input - usually #x# For most functions, the domain is #(-oo, oo)#, the set of all reals. There are a number of factors that can cause this domain to be limited. Here are a few possibilities: In your example, the vertical asymptotes are the cause. When the denominator function #D(x) = 0#, the vertical asymptotes are found to be at #x = +-4# Domain: #(-oo, -4)uu(-4, 4)uu(4, oo)# Find the Range - valid output - usually #y# For most functions, the range is also #(-oo, oo)#, the set of all reals. There are a number of factors that can cause this range to be limited. Here are a few possibilities: In your example, w have a rational function. The degree of the numerator function = 1 #(n = 1)#and the degree of the denominator function = 2, #(m = 2)#. When #n < m# there is a horizontal asymptote at #y = 0#. Range: #(-oo, 0) uu (0, oo)# But you can see from the graph below that the point #(0,0)# exists. This means the domain is actually #(-oo, oo)# How would you know this without graphing the function? Create a table of values. #ul(x|-8" "|-4" "|-2" "|0" "|2" "|4" "|8 " ")# #y|-1/6" "|un" "|1/6" "|0" "|-1/6""|un" "|1/6# #un# = undefined graph{x/(x^2-16) [-10, 10, -5, 5]}

Nathan Axel · Answer

undefined **Domain:** The domain of $g(x) = \frac{x}{x^2 - 16}$ is all real numbers except $x = -4$ and $x = 4$.

**Range:** The range of $g(x)$ is all real numbers except 0.