# 3.2 Domain and range  (Page 2/11)

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Before we begin, let us review the conventions of interval notation:

• The smallest number from the interval is written first.
• The largest number in the interval is written second, following a comma.
• Parentheses, ( or ), are used to signify that an endpoint value is not included, called exclusive.
• Brackets, [ or ], are used to indicate that an endpoint value is included, called inclusive.

See [link] for a summary of interval notation.

## Finding the domain of a function as a set of ordered pairs

Find the domain    of the following function: .

First identify the input values. The input value is the first coordinate in an ordered pair    . There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.

$\left\{2,3,4,5,6\right\}$

Find the domain of the function:

$\left\{\left(-5,4\right),\left(0,0\right),\left(5,-4\right),\left(10,-8\right),\left(15,-12\right)\right\}$

$\left\{-5,\text{\hspace{0.17em}}0,\text{\hspace{0.17em}}5,\text{\hspace{0.17em}}10,\text{\hspace{0.17em}}15\right\}$

Given a function written in equation form, find the domain.

1. Identify the input values.
2. Identify any restrictions on the input and exclude those values from the domain.
3. Write the domain in interval form, if possible.

## Finding the domain of a function

Find the domain of the function $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}-1.$

The input value, shown by the variable $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.

In interval form, the domain of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right).$

Find the domain of the function: $\text{\hspace{0.17em}}f\left(x\right)=5-x+{x}^{3}.$

$\left(-\infty ,\infty \right)$

Given a function written in an equation form that includes a fraction, find the domain.

1. Identify the input values.
2. Identify any restrictions on the input. If there is a denominator in the function’s formula, set the denominator equal to zero and solve for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ . If the function’s formula contains an even root, set the radicand greater than or equal to 0, and then solve.
3. Write the domain in interval form, making sure to exclude any restricted values from the domain.

## Finding the domain of a function involving a denominator

Find the domain    of the function $\text{\hspace{0.17em}}f\left(x\right)=\frac{x+1}{2-x}.$

When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for $\text{\hspace{0.17em}}x.$

$\begin{array}{ccc}\hfill 2-x& =& 0\hfill \\ \hfill -x& =& -2\hfill \\ \hfill x& =& 2\hfill \end{array}$

Now, we will exclude 2 from the domain. The answers are all real numbers where $\text{\hspace{0.17em}}x<2\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}x>2\text{\hspace{0.17em}}$ as shown in [link] . We can use a symbol known as the union, $\text{\hspace{0.17em}}\cup ,$ to combine the two sets. In interval notation, we write the solution: $\left(\mathrm{-\infty },2\right)\cup \left(2,\infty \right).$

Find the domain of the function: $\text{\hspace{0.17em}}f\left(x\right)=\frac{1+4x}{2x-1}.$

$\left(-\infty ,\frac{1}{2}\right)\cup \left(\frac{1}{2},\infty \right)$

Given a function written in equation form including an even root, find the domain.

1. Identify the input values.
2. Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for $\text{\hspace{0.17em}}x.$
3. The solution(s) are the domain of the function. If possible, write the answer in interval form.

## Finding the domain of a function with an even root

Find the domain    of the function $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{7-x}.$

When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.

Set the radicand greater than or equal to zero and solve for $\text{\hspace{0.17em}}x.$

$\begin{array}{ccc}\hfill 7-x& \ge & 0\hfill \\ \hfill -x& \ge & -7\hfill \\ \hfill x& \le & 7\hfill \end{array}$

Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to $\text{\hspace{0.17em}}7,\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left(-\infty ,7\right].$

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