**Domains**

In the beginning of the lesson, we defined the function Eugene as:

*f*(*x*) = √*x*

However, we left out a crucial piece of information: the domain. The domain of this function consisted of only the numbers 1,4,9,25, and 100. Thus, we should have written

*f*(*x*) = √*x* if *x* = 1,4,9,25, or 100

Usually, the domain of a function is not given explicitly like this. In such situations, it is assumed that the domain is as large as it possibly can be. The domain consists of all numbers that don't violate one of the following two fundamental prohibitions:

- Never divide by zero.
- Never take an even root of a negative number.

If you divide by zero, the entire numerical universe will collapse down to a single point. If dividing by zero were allowed, then all numbers would be equal. Four would equal five. Negative and positive would be equivalent. "It's all the same to me" would be the correct answer to every math question. While this might be appealing to some people, it would make calculus, the study of change, impossible. If only one number existed, there could be no change. Thus, we automatically rule out any situation where division by zero might occur.

**Example 1**

What is the domain of

**Solution 1**

We must never let the denominator *x* – 2 be zero, so *x* cannot be 2. Therefore, the domain of this function consists of all real numbers except 2.

The prohibition against even roots (like square roots) of negative numbers is less severe. An even root of a negative number is an imaginary number. Useful mathematics can be done with imaginary numbers. However, for the sake of simplicity, we will avoid them.

**Example 2**

What is the domain of *g*(*x*) = √3*x* + 2?

**Solution 2**

The numbers in the square root must not be negative, so 3*x* + 2 ≥ 0, thus . of all numbers greater than or equal to .

Do note that it is perfectly okay to take the square root of zero, since √0 = 0. It is only when numbers are less than zero that even roots become imaginary.

**Example 3**

Find the domain of .

**Solution 3**

To avoid dividing by zero, we need *x*^{2} + 5*x* + 6 ≠ 0, so (*x* + 3)( *x* + 2) ≠ 0, thus *x* ≠ –3 and *x* ≠ –2. To avoid an even root of a negative number, 4 – *x* 2 ≥ 0, so *x* ≤ 4. Thus, the domain of *k* is *x* ≤ 4, *x* ≠ –3, *x* ≠ –2.

Find practice problems and solutions for these concepts at Calculus Functions Practice Questions

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