Question

What is the domain of the function `f(x)=(x+9)/(x-3)`?

Answer 1

The answer is: `x!=3`, because the denominator cannot be zero.

`dom(f)=(-∞,3)∪(3,+∞)`

Answer 2

The domain of the dependent variable is the spread of the variable on which it depends.
Here, the domain is `x in ( -oo, oo)`, sans x = 3

Explanation:

By actual division,

`f(x) = 1+12/(x-3)`

As `x to 3, y to +-oo`.

As `x to +-oo, y to 1`.

Likewise, the range of f is (-oo, oo)#, sans f = 1.

The lines x = 3 and y =1 are called asymptotes to the graph y = f(x)

graph{y(x-3)-x-9=0 [-80, 80, -40, 40]}