What is the domain and range of # y =1/(2x-4)#?

Question

Ethan Adkins · Accepted Answer

The domain of #y# is #=RR-{2}#
The range of #y#, #=RR-{0}# As you cannot divide by #0#, #2x-4!=0# #x!=2# Therefore, the domain of #y# is #D_y=RR-{2}# To determine the range, we calculate #y^-1# #y=1/(2x-4)# #(2x-4)=1/y# #2x=1/y+4=(1+4y)/y# #x=(1+4y)/(2y)# Thus, #y^-1=(1+4x)/(2x)# The domain of #y^-1# is #D_(y^-1)=RR-{0}# This is the range of #y#, #R_y=RR-{0}# graph{1/(2x-4) [-11.25, 11.25, -5.625, 5.625]}

Eva Adamson · Answer

#"domain " x inRR,x!=2#

#"range " y inRR,y!=0# The denominator of y cannot be zero as this would make y #color(blue)"undefined".#Equating the denominator to zero and solving gives the value that x cannot be. #"solve " 2x-4=0rArrx=2larrcolor(red)" excluded value"# #"domain " x inRR,x!=2# #"to find excluded value/s in the range"# #"Rearrange the function making x the subject"# #rArry(2x-4)=1# #rArr2xy-4y=1# #rArr2xy=1+4y# #rArrx=(1+4y)/(2y)# #"the denominator cannot be zero"# #"solve " 2y=0rArry=0larrcolor(red)" excluded value"# #"range " y inRR,y!=0# graph{1/(2x-4) [-10, 10, -5, 5]}

Avery Alexander · Answer

undefined The domain of the function $ y = \frac{1}{2x - 4} $ is all real numbers except $ x = 2 $ because the denominator cannot be zero. The range is all real numbers except $ y = 0 $ because the function never reaches zero.