Find the domain of the function. (Enter your answer using interval notation.)

Question

f(x) = √4 − x÷ x + 7

Find the domain of the function. (Enter your answer using interval notation.)

Answer 1

If your function is:

then there are two "issues" related to the domain that must be addressed.

1) The domain of the square root function (y = √x) is all numbers greater than or equal to zero (taking the square root of a negative number does not exist in real numbers).

2) The domain of the reciprocal function (y = 1/x) is all real numbers except zero (dividing by zero is not defined).

So the domain of your function must NOT include x + 7 = 0 or x = -7

And the domain of your function must have:

4 - x > 0

-x > -4

x < 4

Putting these in interval notation:

Domain: (-∞, -7) U (-7, 4]

Remember that in interval notation "(" or ")" means that the domain does not include the boundary noted while "[" or "]" means the boundary is included. Also "U" represent the union of the sets.

Answer 2

Hi Alexis -

It's a little unclear what exactly the function is, I will assume that it is

f(x) = sqrt(4-x) / (x+7)

In terms of the elementary functions, some things to remember for these types of problems are the following (assuming we are working with real numbers, as you virtually always are in the first calc sequence):

1. You cannot divide by 0
2. You cannot take an even root of a negative number
3. You cannot take a logarithm of a non-positive number (you cannot apply a log to 0, hence the non-positive phrasing)

While these facts are not exhaustive, you can probably do most of the problems by remembering them.

For this function, we have an even root (the square root is a 1/2 power, remember), and we are dividing:

To apply the square root to 4-x, we must have 4-x ≥ 0, so 4 ≥ x. This is the interval (-∞,4].

We cannot divide by 0, and so we cannot have x+7=0, so x cannot be -7.

Any other values of x will be fine, since then we are not dividing by 0 or taking the square root of a negative number.

Combining the two conditions, we get the domain, in interval notation, to be

(-∞,-7) U (-7,4]