Question

How do you find the domain and range of `y = 3(x-2)/x`?

Answer 1

Domain: `(-oo, 0) uu (0, + oo)`
Range: `(-oo, 3) uu (3, + oo)`

Explanation:

Right from the start, you can say that the domain of the function will not include `x=0`, since that would make the denominator of the fraction equal to zero.

This means that the domain of the function will be `RR - {0}`, or `(-oo, 0) uu (0, + oo)`.

To find if the range of the function has any restrictions, calculate the inverse of `y` by solving for `x`, then switching `x` with `y`

`y = (3x - 6)/x`

`y * x = 3x - 6`

`x (y-3) = 6 implies x = 6/(y-3)`

The inverse function will thus be

`y = 6/(x-3)`

As you can see, this is not defined for `x=3`, which means that your original function cannot take the value `y=3`. The range of the function will thus be `RR- {3}`, or `(-oo, 3) uu (3, + oo)`.

graph{(3(x-2))/x [-10, 10, -5, 5]}