Lesson 3.2: Domain and Range

New Concept

The domain of a function is the set of all valid inputs, while the range is the set of all resulting outputs. This lesson focuses on finding these sets from equations by identifying mathematical restrictions and from graphs.

What’s next

You're about to practice finding the domain and range through interactive examples and worked-out problems. Then, you'll learn to graph functions defined in pieces.

Property

To find the domain of a function, we must identify all input values that are mathematically permitted.
Oftentimes, finding the domain of such functions involves remembering three different forms.
First, if the function has no denominator or an odd 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.

To find the domain from an equation:

1. Identify the input values.
2. Identify any restrictions on the input and exclude those values from the domain.
3. Write the domain in interval form, if possible.

Examples

• For the function $f(x) = x^3 - 2x$, any real number can be an input, so the domain is all real numbers, or $(-\infty, \infty)$.
• For $g(x) = \frac{2x}{x-5}$, the denominator cannot be zero. We set $x-5=0$ and find $x=5$. The domain is all real numbers except 5, written as $(-\infty, 5) \cup (5, \infty)$.
• For $h(x) = \sqrt{x-3}$, the value inside the square root must be non-negative. We set $x-3 \geq 0$, which means $x \geq 3$. The domain is $[3, \infty)$.

Property

Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form $\{x|\,\text{statement about } x\}$ which is read as, “the set of all $x$ such that the statement about $x$ is true.” For example, $\{x|\,4 < x \leq 12\}$.

Interval notation is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. A square bracket $[$ indicates inclusion in the set, and a parenthesis $($ indicates exclusion from the set. For example, $(4, 12]$.

Examples

• The set of all numbers greater than or equal to 8 can be written as $\{x|\,x \geq 8\}$ in set-builder notation or $[8, \infty)$ in interval notation.
• The set of all numbers between -1 and 7, not including either endpoint, is written as $\{x|\,-1 < x < 7\}$ or $(-1, 7)$.
• To represent all real numbers except 4, we use the union symbol: $\{x|\,x \neq 4\}$ or $(-\infty, 4) \cup (4, \infty)$.

Property

Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the $x$-axis.
The range is the set of possible output values, which are shown on the $y$-axis.
Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values.
The domain and range are always written from smaller to larger values.

Examples

• A graph of a line segment starts at $(-4, 2)$ and ends at a hollow circle at $(5, -1)$. The domain is $[-4, 5)$ and the range is $(-1, 2]$.
• For a parabola $f(x) = (x-1)^2 + 3$ with a vertex at $(1, 3)$ that opens upward, the domain is $(-\infty, \infty)$ and the range is $[3, \infty)$.
• A horizontal line at $y=6$ has a domain of $(-\infty, \infty)$ because it extends forever left and right, but its range is just the single value $[6, 6]$ or $\{6\}$.

Explanation

To find a function's domain from its graph, look at how far it spreads horizontally (left to right). For the range, look at its vertical spread (bottom to top). Imagine squashing the graph onto each axis to see its shadow.

Property

A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:

Examples

• Given $f(x) = \begin{cases} x+2 & \text{if } x < 3 \\ x^2 & \text{if } x \geq 3 \end{cases}$. To find $f(1)$, we use the first rule since $1 < 3$, so $f(1) = 1+2 = 3$.
• For the same function, to find $f(4)$, we use the second rule since $4 \geq 3$, so $f(4) = 4^2 = 16$.
• A delivery fee is 5 dollars for orders up to 2 miles and 8 dollars for orders over 2 miles. The function is $C(d) = \begin{cases} 5 & \text{if } 0 < d \leq 2 \\ 8 & \text{if } d > 2 \end{cases}$. The cost for a 1.5-mile delivery is 5 dollars.

Explanation

A piecewise function is a single function that follows different rules for different input values. To use it, first check which 'piece' of the domain your input number belongs to, then apply only that piece's specific formula.