Finding the domain of a function as a set of ordered pairs
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.
Given a function written in equation form, find the domain.
Identify the input values.
Identify any restrictions on the input and exclude those values from the domain.
Write the domain in interval form, if possible.
Finding the domain of a function
Find the domain of the function
$\text{\hspace{0.17em}}f(x)={x}^{2}-1.$
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
$\text{\hspace{0.17em}}\left(-\infty ,\infty \right).$
Given a function written in an equation form that includes a fraction, find the domain.
Identify the input values.
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.
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
$\text{\hspace{0.17em}}f(x)=\frac{x+1}{2-x}.$
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:
$\left(\mathrm{-\infty},2\right)\cup \left(2,\infty \right).$
Given a function written in equation form including an even root, find the domain.
Identify the input values.
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
$\text{\hspace{0.17em}}x.$
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
$\text{\hspace{0.17em}}f(x)=\sqrt{7-x}.$
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
$\text{\hspace{0.17em}}(-\infty ,7].$
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5) and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.