Question

What is the domain and range of `(x-1)/(x-4)`?

Answer 1

Domain: `(-oo, 4) uu (4, + oo)`
Range: `(-oo, 1) uu (1, + oo)`

Explanation:

The domain of the function will include all possible value of `x` except the value that makes the denominator equal to zero. More specifically, `x=4` will be excluded from the domain, which will thus be `(-oo, 4) uu (4, + oo)`.

To determine the range of the function, you can do a little algebraic manipulation to rewrite the function as

`y = ((x - 4) + 3)/(x-4) = 1 + 3/(x-4)`

Since the fraction `3/(x-4)` can never be equal to zero, the function can never take the value

`y = 1 + 0 = 1`

This means that the range of the function will be `(-oo, 1) uu (1, + oo)`.

graph{(x-1)/(x-4) [-18.8, 21.75, -10.3, 9.98]}