DWG NO. 1.1 — Lesson 1 of 6

Functions, Domain, and Range

Unit 1: Functions and Their Graphs · ~20 min

Objective Determine whether a relation is a function, and find the domain and range of a function given its equation or its graph.

What makes a relation a function

A relation is just any pairing of inputs and outputs. A function is a relation with one extra rule: every input produces exactly one output. Nothing stops two different inputs from sharing the same output — that's allowed. What's not allowed is one input leading to two different outputs.

On a graph, this becomes the vertical line test: if any vertical line crosses the graph more than once, it's not a function, because that would mean one x-value maps to more than one y-value.

Domain and range

The domain of a function is the complete set of inputs (x-values) it accepts. The range is the complete set of outputs (y-values) it actually produces. Both are usually written in interval notation.

Watch for zero in a denominator. Any x-value that makes the denominator 0 must be excluded from the domain.
Watch for negatives under an even root. For an even index (square root, fourth root, …), the expression underneath must be ≥ 0.
Everything else is usually all real numbers. Polynomials, for instance, accept any real input.

Graph of f(x) = -(x-1)^2 + 4, a downward-opening parabola with vertex at (1, 4)

f(x) = −(x−1)² + 4 — domain: all real numbers; range: y ≤ 4

Worked Example 1 · Domain with a square root

ProblemFind the domain of f(x) = \(\sqrt{x-3}\).

1A square root needs a non-negative expression underneath: x−3 ≥ 0.

2Solve: x ≥ 3.

Domain: [3, ∞)

Worked Example 2 · Domain with a denominator

ProblemFind the domain of f(x) = \(\frac{1}{x-2}\).

1The denominator cannot equal 0: x−2 ≠ 0, so x ≠ 2.

Domain: (−∞, 2) ∪ (2, ∞)

Worked Example 3 · Domain and range from a graph

ProblemUse the diagram above (f(x) = −(x−1)² + 4) to find the domain and range.

1The graph extends left and right forever, so every x-value is used: domain is all real numbers.

2The graph never rises above y = 4 (the vertex is the highest point), so the range is capped there.

Domain: (−∞, ∞) · Range: (−∞, 4]

Guided practice

1.Is the relation {(1, 2), (2, 4), (1, 5), (3, 6)} a function? Explain why or why not.
Show answer

No. The input 1 appears twice, paired with two different outputs (2 and 5). That violates the one-output-per-input rule, so this relation is not a function.
2.Find the domain of f(x) = \(\sqrt{5-x}\).
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Need 5−x ≥ 0, so x ≤ 5. Domain: (−∞, 5]
3.Find the domain of g(x) = \(\frac{x+1}{x^{2}-9}\).
Show answer

Factor the denominator: x²−9 = (x−3)(x+3), which is 0 when x = 3 or x = −3. Domain: (−∞, −3) ∪ (−3, 3) ∪ (3, ∞)
4.A function's graph is a single unbroken curve that starts at (−2, 0), rises to a peak at (0, 5), and comes back down to (2, 0), going no further left or right. State its domain and range.
Show answer

The curve only exists between x = −2 and x = 2, and only reaches as high as y = 5, never dropping below y = 0. Domain: [−2, 2] · Range: [0, 5]
5.Use the vertical line test to explain why the graph of x = y² (a sideways parabola) is not a function of x.
Show answer

For any positive x-value, a vertical line crosses the sideways parabola twice — once at y = \(\sqrt{x}\) and once at y = −\(\sqrt{x}\). Two outputs for one input means it fails the vertical line test, so x = y² is not a function of x.

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