Exponential and Logarithmic Functions: Background You’ll Need 3
Identify all possible inputs (domain) and outputs (range) for both relations and functions
Domain and Range
We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products.
domain and range
Domain: The domain of a function is the set of all possible input values. These are the values that you can put into the function.
Note that values in the domain are also known as input values, or values of the independent variable, and are often labeled with the lowercase letter [latex]x[/latex].
Range: The range of a function is the set of all possible output values. These are the values that come out of the function.
Values in the range are also known as output values, or values of the dependent variable, and are often labeled with the lowercase letter [latex]y[/latex].
We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [latex][[/latex] when the set includes the endpoint and a parenthesis [latex]([/latex] to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has [latex]$100[/latex] to spend, he or she would need to express the interval that is more than [latex]0[/latex] and less than or equal to [latex]100[/latex] and write [latex]\left(0,\text{ }100\right][/latex].
Before we begin, let us review the conventions of interval notation:
The smallest term from the interval is written first.
The largest term in the interval is written second, following a comma.
Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive.
Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.
Understanding the domain and range helps us to see the full scope of a function and how it operates over different values.
Consider the relation where the input is a family member’s name and the output is their age:
Family Member’s Name (Input)
Family Member’s Age (Output)
Nellie
[latex]13[/latex]
Marcos
[latex]11[/latex]
Esther
[latex]46[/latex]
Samuel
[latex]47[/latex]
Nina
[latex]47[/latex]
Paul
[latex]47[/latex]
Katrina
[latex]21[/latex]
Andrew
[latex]16[/latex]
Maria
[latex]13[/latex]
Ana
[latex]81[/latex]
Domain: The domain is the set of all family members’ names: {Nellie, Marcos, Esther, Samuel, Nina, Paul, Katrina, Andrew, Maria, Ana}
Range: The range is the set of all family members’ ages: [latex]\{13,11,46,47,21,16,81\}[/latex]
Relations can be written as ordered pairs of numbers [latex](x,y)[/latex] or as numbers in a table of values the columns of which each contain inputs or outputs. By examining the inputs ([latex]x[/latex]-coordinates) and outputs ([latex]y[/latex]-coordinates), you can determine whether or not the relation is a function. Remember, in a function, each input corresponds to only one output. That is, each [latex]x[/latex]value corresponds to exactly one [latex]y[/latex] value.
Find the domain of the following function: [latex]\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\}[/latex] .
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.
[latex]\left\{2,3,4,5,6\right\}[/latex]
Let’s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.
How To: 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.
Find the domain of the function [latex]f\left(x\right)={x}^{2}-1[/latex].
The input value, shown by the variable [latex]x[/latex] 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.
[latex]\\[/latex]
In interval form the domain of [latex]f[/latex] is [latex]\left(-\infty ,\infty \right)[/latex].
How To: 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 [latex]x[/latex] . These are the values that cannot be inputs in the function.
Write the domain in interval form, making sure to exclude any restricted values from the domain.
Find the domain of the function [latex]f\left(x\right)=\dfrac{x+1}{2-x}[/latex].
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 [latex]x[/latex].
Now, we will exclude 2 from the domain. The answers are all real numbers where [latex]x<2[/latex] or [latex]x>2[/latex]. We can use a symbol known as the union, [latex]\cup[/latex], to combine the two sets. In interval notation, we write the solution: [latex]\left(\mathrm{-\infty },2\right)\cup \left(2,\infty \right)[/latex].
In interval form, the domain of [latex]f[/latex] is [latex]\left(-\infty ,2\right)\cup \left(2,\infty \right)[/latex].
How To: 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 [latex]x[/latex].
The solution(s) are the domain of the function. If possible, write the answer in interval form.
While zero divided by any number equals zero, division by zero results in an undefined ratio.
An even root of a negative number does not exist in the real numbers.[latex]\sqrt{-1} = i[/latex]Since the domain of any function defined in the real plane is the set of all real input into the function, we must exclude any values of the input variable that create undefined expressions or even roots of a negative.
Find the domain of the function [latex]f\left(x\right)=\sqrt{7-x}[/latex].
When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.
[latex]\\[/latex]
Set the radicand greater than or equal to zero and solve for [latex]x[/latex].
Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to [latex]7[/latex], or [latex]\left(-\infty ,7\right][/latex].
How To: Given the formula for a function, determine the domain and range.
Exclude from the domain any input values that result in division by zero.
Exclude from the domain any input values that have nonreal (or undefined) number outputs.
Use the valid input values to determine the range of the output values.
Look at the function graph and table values to confirm the actual function behavior.
Find the domain and range of [latex]f\left(x\right)=2{x}^{3}-x[/latex].
There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.
[latex]\\[/latex]
The domain is [latex]\left(-\infty ,\infty \right)[/latex] and the range is also [latex]\left(-\infty ,\infty \right)[/latex].
Find the domain and range of [latex]f\left(x\right)=2\sqrt{x+4}[/latex].
We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.
[latex]x+4\ge 0\text{ when }x\ge -4[/latex]
The domain of [latex]f\left(x\right)[/latex] is [latex]\left[-4,\infty \right)[/latex].We then find the range. We know that [latex]f\left(-4\right)=0[/latex], and the function value increases as [latex]x[/latex] increases without any upper limit. We conclude that the range of [latex]f[/latex] is [latex]\left[0,\infty \right)[/latex].
[latex]\\[/latex] Analysis of the Solution [latex]\\[/latex] The graph below represents the function [latex]f[/latex].
