How do you find the domain and range of `f(x) = sqrt(4+x) / (1-x)`?

The domain of the function represents all the values that `x` can take for which `f(x)` is defined.

Right from the start, you should be able to say that the domain of the function cannot include `x = 1` because that would make the function undefined.

`1 - x != 0 implies x != 1`

Moreover, notice that the function contains the square root of an expression that depends on the value of `x`. As you know, when working with real numbers, you cannot take the square root of a negative number.

This implies that you need

`4 + x >= 0 implies x >= - 4`

Therefore, you can say that the domain of the function will be

`x in [-4, 1) uu (1, +oo)`

THis tells you that the function is defined for any value of `x` that it greater than or equal to `-4` and is not equal to `1`.

The range of the function tells you all the possible values that `f(x)` can take for valid domain values.

In this case, the square root of a positive number will produce a positive number, which means that regardless what value of `x` you plug in from the domain of the function, you will have

`sqrt(4 -x ) >= 0`

Now, for any value of `x in [-4, 1)`, you will have

`{( sqrt(4 + x) >= 0), (1 - x > 0) :} implies f(x) >= 0`

and for any value of `x in (1, +oo)`, you will have

`{( sqrt(4 + x) > 0), (1 - x < 0) :} implies f(x) < 0`

This means that the range of the function is

`(-oo, 0] uu (0, +oo) = (- oo, +oo)`

graph{sqrt(4+x)/(1-x) [-10, 10, -5, 5]}