Domain and Range Analysis

When working with functions, it is crucial to know what values can go into the function and what values can come out. These sets of values are called the domain and range.

• Domain: The set of all possible input values ($x$-values) for which the function is defined.
• Range: The set of all possible output values ($y$-values) the function can produce.

We typically express these sets using interval notation, where brackets [ ] mean a value is included, and parentheses ( ) mean a value is excluded (or used for infinity).

Finding Domain from an Equation

When looking at an algebraic equation, the domain is usually all real numbers, $(-\infty, \infty)$, unless there are specific mathematical restrictions. The two most common restrictions you will encounter are:

1. Division by zero: The denominator of a fraction cannot be zero.
2. Square roots of negative numbers: The expression inside an even root (like a square root) must be greater than or equal to zero.

Example:

Find the domain of $f(x) = \frac{\sqrt{x - 2}}{x - 5}$.

Step 1: Check the square root. The expression under the radical must be non-negative: $x - 2 \ge 0 \implies x \ge 2$

Step 2: Check the denominator. The denominator cannot be zero: $x - 5 \neq 0 \implies x \neq 5$

Step 3: Combine and write in interval notation. The $x$-values must be at least $2$, but cannot equal $5$. In interval notation, the domain is: $[2, 5) \cup (5, \infty)$

Domain and Range from a Graph

Graphs provide a visual way to determine both domain and range:

• Domain: Scan the graph horizontally from left to right to see all the $x$-values the graph covers.
• Range: Scan the graph vertically from bottom to top to see all the $y$-values.

Example:

Imagine a graph of a parabola that opens upwards, with its lowest point (vertex) at $(1, -3)$.

• Domain: The arrows point outward to the left and right infinitely. The domain is $(-\infty, \infty)$.
• Range: The lowest $y$-value is $-3$ (which is included), and the graph goes up infinitely. The range is $[-3, \infty)$.

Real-World Contexts

In word problems, the domain and range are often restricted by physical reality and logic. For example, if a function calculates the height of a thrown ball over time $t$, the domain (time) cannot be negative ($t \ge 0$), and the range (height) cannot drop below ground level ($y \ge 0$). Always check if your mathematical answers make sense in the context of the real world!