How do you find the domain and range of #f(x) =(x+4)/(x+2)#?

Question

Abigail Allen · Accepted Answer

#x inRR,x!=-2#
#y inRR,y!=1# The denominator of f(x) cannot be zero as this would make f(x) #color(blue)"undefined".#Eqating the denominator to zero and solving gives the value that x cannot be. #"solve " x+2=0rArrx=-2larrcolor(red)" excluded value"# #rArr"domain is " x inRR,x!=-2# #"to find any excluded values in the range"# #"rearrange f(x) = y making x the subject"# #rArry=(x+4)/(x+2)larr" cross-multiply"# #rArry(x+2)=x+4# #rArrxy+2y-x=4# #rArrx(y-1)=4-2y# #rArrx=(4-2y)/(y-1)# #"the denominator cannot be zero"# #"solve "y-1=0rArry=1larrcolor(red)" excluded value"# #"range is " y inRR,y!=1#

Eli Assouline · Answer

undefined To find the domain of the function $ f(x) = \frac{x + 4}{x + 2} $, set the denominator $ x + 2 $ not equal to zero and solve for $ x $. The domain is all real numbers except for $ x = -2 $.

To find the range, observe that as $ x $ approaches positive or negative infinity, $ f(x) $ approaches 1. Also, as $ x $ approaches -2 from either side, $ f(x) $ approaches positive or negative infinity. Therefore, the range of the function is all real numbers except for zero.