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 following 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.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 following 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 [latex]0[/latex] and solve for [latex]x[/latex].
Now, we will exclude [latex]2[/latex] 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 following 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. Set the radicand greater than or equal to zero and solve for [latex]x[/latex].
Now, we will exclude any number greater than [latex]7[/latex] from the domain. The answers are all real numbers less than or equal to [latex]7[/latex], or [latex]\left(-\infty ,7\right][/latex].
Can there be functions in which the domain and range do not intersect at all?
Yes. For example, the function [latex]f\left(x\right)=-\frac{1}{\sqrt{x}}[/latex] has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function’s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.
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 the following:
[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. 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 the following:
[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].
