What is the range of the function #(x-1)/(x-4)#?

2 Answers
Jun 2, 2017

Answer:

The range of #(x-1)/(x-4)# is #RR"\"{1}# a.k.a. #(-oo, 1) uu (1, oo)#

Explanation:

Let:

#y = (x-1)/(x-4) = (x-4+3)/(x-4) = 1+3/(x-4)#

Then:

#y - 1 = 3/(x-4)#

Hence:

#x-4 = 3/(y-1)#

Adding #4# to both sides, we get:

#x = 4+3/(y-1)#

All these steps are reversible, except division by #(y-1)#, which is reversible unless #y=1#.

So given any value of #y# apart from #1#, there is a value of #x# such that:

#y = (x-1)/(x-4)#

That is, the range of #(x-1)/(x-4)# is #RR"\"{1}# a.k.a. #(-oo, 1) uu (1, oo)#

Here's the graph of our function with its horizontal asymptote #y=1#

graph{(y-(x-1)/(x-4))(y-1) = 0 [-5.67, 14.33, -4.64, 5.36]}

If the graphing tool allowed, I would also plot the vertical asymptote #x=4#

Jun 2, 2017

Answer:

#y inRR,y!=1#

Explanation:

#"rearrange "y=(x-1)/(x-4)" making x the subject"#

#rArry(x-4)=x-1larrcolor(blue)" cross-multiplying"#

#rArrxy-4y=x-1#

#rArrxy-x=-1+4y#

#rArrx(y-1)=4y-1#

#rArrx=(4y-1)/(y-1)#

#"the denominator of x cannot be zero as this would make"#
#"x undefined."#

#"equating the denominator to zero and solving gives the"#
#"value that y cannot be"#

#"solve " y-1=0rArry=1larrcolor(red)" excluded value"#

#rArr"range is " y inRR,y!=1#