Question

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

Answer 1

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)`.