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

1 Answer
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).