Identifying a Function from Ordered Pairs and Graphs

A function gives one output for each input. Different inputs may share an output, but one input cannot point to two different outputs. This definition can be checked in a table, a set of ordered pairs, a mapping diagram, an equation, or a graph. Identify which value is the input before deciding whether the relation qualifies.

Check ordered pairs for repeated inputs

In a pair (x, y), the first coordinate x is the input and the second coordinate y is the output when the relation is written as y depending on x. Scan the first coordinates. If the same x appears twice with different y-values, the relation is not a function. If a repeated x maps to the same y again, it still has only one output and does not violate the definition.

For example, {(1, 4), (2, 5), (3, 5)} is a function. Inputs 1, 2, and 3 each have one output; inputs 2 and 3 happen to share output 5, which is allowed. The relation {(1, 4), (2, 5), (1, 7)} is not a function because input 1 is paired with output 4 and output 7.

Order matters. A relation may fail to be a function of x but still be a function of y. In {(1, 4), (2, 4)}, each x has one y, so y is a function of x. But y = 4 is associated with two different x-values, so x is not a function of y. Read the question’s direction rather than treating “function” as a property independent of which variable is input.

Use the vertical line test on a graph

For a graph of y as a function of x, imagine sliding a vertical line from left to right. If a vertical line touches the graph at two or more points, the same x-coordinate has multiple y-coordinates. The graph does not represent y as a function of x. If every vertical line touches at most one point, it passes the vertical line test.

A straight nonvertical line passes the test: each x-coordinate has one point on it. A vertical line such as x = 3 fails because every point on that line has the same x-value and a different y-value. A circle also fails for its usual full graph; a vertical line through its middle meets the circle at two points. The test does not require calculating coordinates for every point.

Common graph examples

The graph y = x² is a function. For every chosen x there is one squared value. Its outputs are never negative, but that does not matter for the function test. By contrast, the circle x² + y² = 25 fails as y in terms of x: for many x-values, there is an upper and lower y-value. For instance, x = 0 gives y = 5 and y = −5.

A sideways parabola such as x = y² also fails as y a function of x over its full graph because positive x-values can correspond to two y-values, one positive and one negative. A graph restricted to only its upper half could pass; a restriction changes the relation. Always test the graph or domain actually shown.

Function and one-to-one are different tests

A function allows different inputs to share the same output. A one-to-one function adds the stricter condition that different inputs must have different outputs. For example, f(x) = x² is a function, but it is not one-to-one over all real numbers because f(2) = 4 and f(−2) = 4. Do not use the horizontal line test to decide whether a relation is a function; vertical lines test functions, while horizontal lines test whether a function is one-to-one.

This distinction explains why a graph may pass the vertical line test but fail the horizontal line test. It still is a function; it simply repeats some output values. One-to-one matters for inverse functions and other advanced topics, while the basic function test is about one input having no more than one output.

Check a table or mapping diagram

In a table, check the input column. If input 3 appears in two rows with outputs 8 and 10, it is not a function. If 3 appears twice with output 8 both times, duplicate rows do not create multiple outputs. In a mapping diagram, each input should have exactly one arrow leaving it. Several arrows may arrive at the same output.

A missing input or blank output may make the relation incomplete rather than a function, depending on how the domain is defined. A function must assign an output to every input in its domain. A question may define a relation using only the listed pairs, or may ask whether a rule defines a function for all allowed real inputs. Use the stated domain and do not silently assume unlisted values are included.

Read equations with care

An equation in y = f(x) form gives one output for each allowed input if the expression is defined there. But an equation involving both x and y might not define a function of x. For x² + y² = 25, solving for y gives y = ±√(25 − x²), generally two outputs for interior x-values. The plus-minus sign exposes the failure without drawing a graph.

An equation can still define a function after restricting its domain or selecting one branch. For the circle’s upper half, y = √(25 − x²) on −5 ≤ x ≤ 5 gives one nonnegative output for every allowed x. Read any domain restriction or instruction that says “upper semicircle” before classifying.

A reliable decision sequence

1. Identify which variable is the input and which is the output.
2. For ordered pairs or tables, scan for one input paired with different outputs.
3. For a graph, use vertical lines and check whether any one intersects the graph more than once.
4. For an equation, solve or reason about how many outputs are possible for a fixed input.
5. Check the stated domain or any restriction that may remove a second output.
6. Do not confuse passing the function test with being one-to-one; that uses a separate horizontal line test.

Common mistakes

• Treating a repeated output as a violation. Functions may send many inputs to the same output.
• Looking for repeated y-values instead of checking whether an x-value has two different y-values.
• Reversing the input-output direction in an ordered pair.
• Using a horizontal line instead of a vertical line to test whether a graph is a function.
• Assuming every equation with x and y defines y as a function of x.
• Ignoring a restricted domain or selecting both branches of a square-root equation when the graph shows only one.
• Thinking a relation must be one-to-one to count as a function.

Exam takeaway

A function assigns one output to each allowed input. For ordered pairs, check whether any input is matched to different outputs. For a graph, use the vertical line test. Repeated outputs are allowed; repeated inputs with different outputs are not. Keep the direction, domain, and distinction from one-to-one clear.