3.2 Domain and Range

Domain and Range of Functions

The domain of a function is the set of all inputs you can plug in. The range is the set of all outputs the function can produce. Knowing how to find both is essential for graphing, solving equations, and working with functions in applied problems.

Domain Restrictions on Functions

The domain of a function includes every real number unless something in the function's structure forces you to exclude certain values. There are three main situations that create restrictions.

Division by zero. If a function has a variable in the denominator, any input that makes the denominator equal zero must be excluded.

• $f(x) = \frac{1}{x - 2}$ → Setting $x - 2 = 0$ gives $x = 2$
• Domain: $\{x \mid x \neq 2\}$ or $(-\infty, 2) \cup (2, \infty)$

Even roots. Square roots, fourth roots, and other even roots require the expression underneath to be greater than or equal to zero (you can't take the square root of a negative number in the reals).

• $f(x) = \sqrt{x + 1}$ → Set $x + 1 \geq 0$, so $x \geq -1$
• Domain: $\{x \mid x \geq -1\}$ or $[-1, \infty)$

Logarithms. The argument of any logarithm (the expression inside) must be strictly positive.

• $f(x) = \ln(x - 3)$ → Set $x - 3 > 0$, so $x > 3$
• Domain: $\{x \mid x > 3\}$ or $(3, \infty)$

When a function combines more than one of these features, find each restriction separately, then take the intersection of the allowed values.

You can express domains using set-builder notation like $\{x \mid x \in \mathbb{R}\}$ or interval notation like $(-\infty, \infty)$. Parentheses mean the endpoint is excluded; brackets mean it's included.

Analysis of Piecewise Functions

A piecewise function is built from two or more sub-functions, each applying on a different interval of the domain. Here's how to work with them:

1. Identify each piece and the interval where it applies.
2. Find the domain of each piece by looking at its interval and any algebraic restrictions within that piece.
3. Combine the intervals to get the overall domain. Check for gaps or overlaps.

Consider this example:

$f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 2x - 1 & \text{if } x \geq 1 \end{cases}$

• The first piece, $x^2$, covers $(-\infty, 1)$.
• The second piece, $2x - 1$, covers $[1, \infty)$.
• Together, the domain is $(-\infty, \infty)$ with no gaps.

Graphing tips:

• Graph each sub-function only within its stated interval.
• At boundary points, use a closed circle (●) if the endpoint is included ($\leq$ or $\geq$) and an open circle (○) if it's excluded ($<$ or $>$).
• In the example above, the point at $x = 1$ gets an open circle on the parabola and a closed circle on the line, since the second piece uses $\geq$.

To find the range of a piecewise function, look at the output values each piece produces over its interval, then combine them.

Real-World Applications of Domain and Range

In applied problems, the context itself often restricts the domain and range to values that make physical sense.

Example 1: Car rental cost. A company charges a $50 base fee plus $0.25 per mile driven. The cost function is $C(m) = 50 + 0.25m$.

• Domain: $[0, \infty)$ because miles driven can't be negative.
• Range: $[50, \infty)$ because the minimum cost is the $50 base fee (when $m = 0$).

Example 2: Garden dimensions. A rectangular garden has a fixed perimeter of 60 feet. If the width is $w$, then the length is $\frac{60 - 2w}{2} = 30 - w$.

• Domain: $(0, 30)$ because the width must be positive and less than 30 (otherwise the length would be zero or negative).
• Range: $(0, 30)$ for the same reason, since length and width are symmetric here.

Notice how both examples use interval notation. In context problems, always ask: What input values are physically meaningful? and What output values can the function actually produce?

Advanced Function Concepts

These ideas build directly on domain and range and will come up in later sections:

• Function composition combines two functions by using the output of one as the input of the other. Written $f(g(x))$, the domain is limited to inputs where $g(x)$ is defined and where $g(x)$ falls within the domain of $f$.
• Inverse functions reverse the input-output relationship. If $f(a) = b$, then $f^{-1}(b) = a$. The domain of $f$ becomes the range of $f^{-1}$, and vice versa.
• One-to-one functions pass the horizontal line test: every output corresponds to exactly one input. Only one-to-one functions have inverses that are also functions.