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Before we begin, let us review the conventions of interval notation:
See [link] for a summary of interval notation.
Find the domain of the following function: $\text{\hspace{0.17em}}\left\{\left(2,\text{}10\right),\left(3,\text{}10\right),\left(4,\text{}20\right),\left(5,\text{}30\right),\left(6,\text{}40\right)\right\}$ .
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.
Find the domain of the function:
$\left\{(\mathrm{-5},4),(0,0),(5,\mathrm{-4}),(10,\mathrm{-8}),(15,\mathrm{-12})\right\}$
$\{-5,\text{\hspace{0.17em}}0,\text{\hspace{0.17em}}5,\text{\hspace{0.17em}}10,\text{\hspace{0.17em}}15\}$
Given a function written in equation form, find the domain.
Find the domain of the function $\text{\hspace{0.17em}}f(x)={x}^{2}-1.$
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: $\text{\hspace{0.17em}}f(x)=5-x+{x}^{3}.$
$\left(-\infty ,\infty \right)$
Given a function written in an equation form that includes a fraction, find the domain.
Find the domain of the function $\text{\hspace{0.17em}}f(x)=\frac{x+1}{2-x}.$
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 $\text{\hspace{0.17em}}x.$
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: $\text{\hspace{0.17em}}f(x)=\frac{1+4x}{2x-1}.$
$\left(-\infty ,\frac{1}{2}\right)\cup \left(\frac{1}{2},\infty \right)$
Given a function written in equation form including an even root, find the domain.
Find the domain of the function $\text{\hspace{0.17em}}f(x)=\sqrt{7-x}.$
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 $\text{\hspace{0.17em}}x.$
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 $\text{\hspace{0.17em}}(-\infty ,7].$
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