What is the difference between domain and range

The domain tells you which x‑values you are allowed to plug into a formula without breaking mathematical rules such as division by zero, taking an even root of a negative number, or applying a logarithm to a non‑positive argument. The range tells you what y‑values actually appear after the function processes every allowed x. Knowing both sets lets you sketch accurate graphs, solve equations, and avoid illegal operations.

Visualizing Domain and Range

On a Cartesian graph the domain appears as the horizontal stretch of the curve, while the range appears as the vertical stretch. For the parabola f(x)=$x^2$ the curve extends infinitely left and right, so the domain is all real numbers, but it never dips below the x‑axis, giving a range of non‑negative numbers. If a graph stops at a vertical line, that line marks the boundary of the domain; if it stops at a horizontal line, that marks the boundary of the range.

Remember these facts:

• Domain is read left‑to‑right on the x‑axis.
• Range is read bottom‑to‑top on the y‑axis.
• Vertical‑line test checks if a relation is a function, horizontal‑line test checks the range.
• Endpoints may be included or excluded depending on inequalities.

To determine domain and range:

1. 1Identify any restrictions on x such as division by zero, even roots, or logarithms.
2. 2Solve the restrictions and write the domain in interval notation.
3. 3Sketch or analyze the graph to locate the smallest and largest y‑values.
4. 4Write the range in interval notation, noting open or closed ends.

Example: f(x)=$\sqrt{x-2}$. Step 1: the square‑root requires its argument to be non‑negative, so x-2\ge0, giving x\ge2. Thus the domain is [2,\infty). Step 2: for every x in the domain, the output is a non‑negative square root, so y\ge0. The smallest output occurs at x=2, where y=0, and y grows without bound as x increases. Therefore the range is [0,\infty). This concrete walk‑through shows how the domain and range are derived from the same expression but are not identical.

Domain and range for common functions: