9.4: The Domain and Range of a Function
- Page ID
- 174216
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Lecture: A Detailed Overview of the Concept of Domain
Lecture: Domains of Graphs of Functions
Example: Finding the Domain and Range of a Given Graph
Examples: Finding the Domain of a Rational Function
Example #1
Example #2
Example: The General Idea of How to Determine the Domain of a Radical Function
Example: Finding the Domain of a Function that is a Ratio of Square Roots
Definitions and Theorems
The domain of a function \(f\) is the set of all input values \(x\) for which \(f(x)\) is defined.
The range (or image) of a function \(f\) is the set of all output values the function attains—that is, the set of all values \(f(x)\) as \(x\) varies over the domain.
The domain of the function \( y = f(x) \) is the set of all values of \( x \) for which the function is defined and can be evaluated. The range of the function is the set of all outputs of the function corresponding to the values of \( x \) in the domain.
Throughout the Codex, range means the set of output values a function actually attains. Some sources reserve the word for a larger set of allowable outputs; read it here only in the sense given above.
When a function is specified only by a formula, with no domain stated, its natural domain (or implied domain) is the set of all real numbers for which the formula produces a real-number output.
Let \(f\) be a real-valued function graphed in the Cartesian plane. The domain of \(f\) is the set of all \(x\)-coordinates of points on the graph—the projection of the graph onto the \(x\)-axis. The range of \(f\) is the set of all \(y\)-coordinates of points on the graph—the projection of the graph onto the \(y\)-axis.
To determine the natural domain of a real-valued function given by a formula, exclude every input that produces division by zero in a denominator, and every input that produces a negative number beneath a radical of even index. Every remaining real number belongs to the natural domain.
When a formula carries more than one restriction, the natural domain is the set of inputs that satisfy every restriction at once; take the intersection of the individual requirements. In particular, an even radical that appears in a denominator must have a strictly positive radicand, since the expression can be neither negative (undefined) nor zero (division by zero).
Examples
Find the domain of \(f(x)=2x^2-3x+1\).
- Solution
- The formula is a polynomial: it contains no denominator and no radical, so no input can produce division by zero or an even root of a negative number. Every real number yields a real output. The domain is \(\mathbb{R}\), written in interval notation as \((-\infty,\infty)\).
Find the domain of \(f(x)=\dfrac{2x-1}{x^2-x-6}\).
- Solution
-
Only division by zero can restrict the domain, so locate the inputs that make the denominator zero. Factoring the denominator,\[x^2-x-6=(x-3)(x+2).\nonumber\]The denominator is zero at \(x=3\) and \(x=-2\), so both values are excluded; every other real number is permitted. The domain is\[(-\infty,-2)\cup(-2,3)\cup(3,\infty).\nonumber\]
Find the domain of \(f(x)=\sqrt{2x-8}\).
- Solution
-
The radical has even index, so its radicand must be nonnegative:\[2x-8\geq 0.\nonumber\]Solving gives \(x\geq 4\). The domain is \([4,\infty)\).
Find the domain of \(f(x)=\dfrac{\sqrt{x+5}}{x-2}\).
- Solution
-
Two restrictions apply. The even radical requires a nonnegative radicand,\[x+5\geq 0,\nonumber\]so \(x\geq -5\). The denominator requires \(x-2\neq 0\), so \(x\neq 2\). Take the intersection: begin with \([-5,\infty)\) and remove the single value \(x=2\), which lies inside that interval. The domain is\[[-5,2)\cup(2,\infty).\nonumber\]
Find the range of \(f(x)=3-(x-1)^2\).
- Solution
-
For every real \(x\), the square \((x-1)^2\) is nonnegative, so \((x-1)^2\geq 0\). Multiplying by \(-1\) reverses the inequality, giving \(-(x-1)^2\leq 0\), and adding \(3\) yields\[f(x)=3-(x-1)^2\leq 3.\nonumber\]The value \(3\) is attained at \(x=1\), and \(f(x)\) decreases without bound as \(x\) moves away from \(1\). Hence the outputs fill every value up to and including \(3\), and the range is \((-\infty,3]\).
The graph of a function \(f\) is shown in Figure \(\PageIndex{1}\). Determine the domain and range of \(f\).
- Solution
-
Project the graph onto the \(x\)-axis. The curve spans horizontally from \(x=-2\) to \(x=2\), and both endpoints are included, so the domain is \([-2,2]\).
Project the graph onto the \(y\)-axis. The outputs run from \(y=0\) at the two endpoints up to \(y=2\) at the top of the curve, with every value in between attained, so the range is \([0,2]\).
MyOpenMath Practice
Sources
Several parts of this text use modifications from the following sources:
- Wikipedia article: "Domain of a function"
- Wikipedia article: "Range of a function"
All of these sources are released under the Creative Commons Attribution-Share-Alike License 4.0.