What is the domain and range of #y= 1 / (x-3) #?

1 Answer
Adam L.
Mar 20, 2017

Domain: #RR-{3}#, or #(-oo, 3) uu (3, oo)#
Range: #RR-{0}#, or #(-oo, 0) uu (0, oo)#

Explanation:

You can't divide by zero, meaning the denominator of the fraction cannot be zero, so
#x-3!=0#
#x!=3#
Thus, the domain of the equation is #RR-{3}#, or #(-oo, 3) uu (3, oo)#
Alternately, to find the domain and range, look at a graph:
graph{1/(x-3) [-10, 10, -5, 5]}
As you can see, the x never equals 3, there is a gap at that point, so the domain doesn't include 3 - and there is a vertical gap in the range of the graph at y=0, so the range doesn't include 0.
So, again, the domain is #RR-{3}#, or #(-oo, 3) uu (3, oo)#
And the range is #RR-{0}#, or #(-oo, 0) uu (0, oo)#.

NOTE: Another way to find y that may or may not be allowed (solving for x):
Multiply both sides by x:
#y(x-3)=1#
Divide by y:
#x-3=1/y#
Add 3:
#x=1/y+3#
Since you can't divide by zero, #y!=0#, and the range of y is #RR-{0}# or #(-oo, 0) uu (0, oo)#.

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