Which Relation Graphed Below Is A Function

Which relation graphed below is a function?
Determining whether a graph represents a function is a fundamental skill in algebra and calculus. By applying the vertical line test and understanding the definition of a function, you can quickly decide if each plotted relation meets the criteria. This guide walks you through the concept, provides step‑by‑step instructions, illustrates with examples, highlights common pitfalls, and answers frequently asked questions so you can confidently identify functions from any graph.

Introduction

A relation is any set of ordered pairs ((x, y)). A function is a special type of relation where every input (x) is paired with exactly one output (y). When a relation is displayed on a coordinate plane, the question “which relation graphed below is a function?” can be answered visually using the vertical line test. If any vertical line intersects the graph more than once, the relation fails to be a function; if every vertical line touches the graph at most once, it passes the test and is a function.

Understanding Relations and Functions

Definition of a Function

Formal definition: A function (f) from a set (A) (domain) to a set (B) (codomain) assigns each element (x \in A) a unique element (y \in B).
Everyday wording: For each (x) value there is only one corresponding (y) value.

Why the Vertical Line Test Works

A vertical line has the equation (x = c), where (c) is a constant. If the line crosses the graph at two different points ((c, y_1)) and ((c, y_2)) with (y_1 \neq y_2), then the same input (x = c) yields two different outputs, violating the function rule. Conversely, if no vertical line produces more than one intersection, each (x) maps to a single (y).

The Vertical Line Test: Step‑by‑Step Guide

Follow these steps to evaluate any graph:

Imagine or draw a vertical line (a line parallel to the (y)-axis).
Slide the line left to right across the entire width of the graph.
Observe the intersections:
- If the line touches the graph zero or one time at every position, the relation is a function.
- If the line touches the graph two or more times at any position, the relation is not a function.
Note edge cases:
- Graphs with gaps or holes still obey the test as long as no vertical line hits more than one point.
- Points that are isolated (a single dot) are fine; they represent a single (y) for that (x). ### Quick Checklist

✅ Passes → each (x) → one (y) → function.
❌ Fails → any (x) → multiple (y) → not a function.

Examples of Graphs That Are Functions

Graph Type	Description	Why It Passes the Vertical Line Test
Linear function (y = 2x + 3)	Straight line, non‑vertical	Any vertical line cuts the line exactly once.
Quadratic parabola (y = x^2)	U‑shaped curve opening upward	Vertical lines intersect the parabola at most once (except at the vertex where they touch once).
Cubic function (y = x^3 - x)	S‑shaped curve	Each vertical line meets the curve once; the curve never doubles back horizontally.
Square root function (y = \sqrt{x}) (domain (x \ge 0))	Half‑parabola lying on its side	For each (x) there is a single non‑negative (y); vertical lines hit the curve once or not at all.
Piecewise function (e.g., (f(x) = \begin{cases} -x & x < 0 \ x^2 & x \ge 0 \end{cases}))	Two different rules joined at (x=0)	At the join point the graph is continuous and still yields a single (y) for (x=0).

Visual tip: If you can draw the graph without lifting your pen and never retrace horizontally, it’s likely a function.

Examples of Graphs That Are Not Functions | Graph Type | Description | Why It Fails the Vertical Line Test |

|------------|-------------|--------------------------------------| | Circle (x^2 + y^2 = r^2) | Round shape centered at origin | A vertical line through the center hits the circle at two points (top and bottom). | | Ellipse (\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1) | Stretched circle | Same reasoning as the circle; central vertical lines intersect twice. | | Sideways parabola (x = y^2) | Parabola opening to the right | Vertical lines intersect the curve twice (except at the vertex where they touch once). | | Figure‑eight (lemniscate) | Two loops crossing at the origin | Vertical lines through the crossing point meet the graph four times. | | Vertical line (x = 4) | Straight line parallel to (y)-axis | Any vertical line coincides with the graph, yielding infinitely many points for the same (x). |

Note: Relations that fail the test can still be useful (e.g., circles in geometry), but they do not satisfy the strict definition of a function.

Common Mistakes and Tips

Mistake 1: Confusing the Horizontal Line Test with the Vertical Line Test

Horizontal line test determines if a function is one‑to‑one (injective). It does not tell you whether a relation is a function.
Tip: Remember “vertical” for input uniqueness; “horizontal” for output uniqueness.

Mistake 2: Overlooking Discontinuities

A graph with a hole (open circle) still counts as a single point for that (x) if the hole is the only representation.
**

If there are multiple filled points at the same (x), it fails the test.

Mistake 3: Assuming All Curves Are Functions

Many familiar shapes (circles, ellipses, sideways parabolas) are not functions.
Tip: Sketch the graph mentally or on paper and try a few vertical lines before concluding.

Mistake 4: Ignoring Domain Restrictions

Some relations are functions only on a restricted domain (e.g., (y = \sqrt{x}) is a function for (x \ge 0) but not for all real (x)).
Tip: Always check the stated domain before applying the test.

Quick Checklist

Identify the domain of the relation.
Draw or imagine vertical lines across the entire domain.
Count intersections for each line.
If any line meets the graph more than once, it’s not a function.

Conclusion

The vertical line test is a simple yet powerful visual tool for determining whether a relation defines a function. By ensuring that each input (x) corresponds to exactly one output (y), it upholds the fundamental definition of a function. While it’s easy to confuse this test with the horizontal line test or to overlook domain restrictions, careful application—paired with an awareness of common pitfalls—will help you accurately classify graphs. Whether you’re sketching parabolas, circles, or more complex curves, remembering to “check vertically” will guide you in distinguishing functions from non-functions and deepen your understanding of mathematical relationships.

Beyond the basic visual check, the vertical line test can be extended to more abstract settings and combined with other analytical tools to deepen our insight into relations.

Applying the Test to Piecewise Definitions
When a relation is described by different formulas on separate intervals, each piece must individually satisfy the test. For instance, the relation

[ y=\begin{cases} x^2 & \text{if } x<0,\ \sqrt{x} & \text{if } x\ge 0, \end{cases} ]

passes the vertical line test because every vertical line intersects either the left‑hand parabola or the right‑hand root curve, but never both at the same (x). If the pieces overlapped in (x)‑range and produced conflicting (y)-values, the test would fail, signalling that the piecewise rule does not define a function.

Implicit Relations and the Test
Some curves are given implicitly, such as (x^2+y^2=1) (a circle). Solving for (y) yields two branches, (y=\pm\sqrt{1-x^2}). A vertical line at (x=0.5) meets the circle at two points, confirming that the implicit relation is not a function. However, if we restrict to the upper semicircle by adding the condition (y\ge0), the implicit equation together with the inequality does define a function, and the vertical line test now succeeds.

Parametric Curves
A parametric representation ((x(t),y(t))) can trace a shape that fails the vertical line test even though each coordinate is a function of the parameter. For example, the lemniscate (x(t)=\frac{\sin t}{1+\cos^2 t},; y(t)=\frac{\sin t\cos t}{1+\cos^2 t}) loops back on itself. While (x(t)) and (y(t)) are each functions of (t), the resulting curve in the (xy)-plane fails the test because a single (x)-value can correspond to two different (t)-values, producing two distinct (y)-points. This illustrates that the vertical line test assesses the graph of a relation, not the functional nature of its parameterization.

Using Technology
Graphing calculators and software often include a “vertical line” feature that lets you drag a line across the display and instantly see the number of intersections. This interactive approach reinforces the intuition behind the test and helps catch subtle issues such as isolated points or removable discontinuities that might be missed in a static sketch.

Limitations of the Test
The vertical line test works only for relations expressed as subsets of the Cartesian plane. In higher dimensions, analogous checks involve hyperplanes: a relation (F(x_1,\dots,x_n,y)=0) defines (y) as a function of the other variables iff every ((x_1,\dots,x_n))-plane intersects the graph in at most one point. Recognizing this generalization prepares students for multivariable calculus and implicit function theorems.

Practice Problems to Solidify Understanding

Sketch (y^2 = x) and apply the vertical line test.
Determine whether the relation defined by (xy = 1) is a function.
For the piecewise rule (y = \begin{cases} 2x+1 & x\le 3 \ -x+5 & x>3 \end{cases}), verify the test on each interval and at the boundary (x=3). Working through these examples reinforces the habit of scanning the entire domain, checking for overlapping pieces

Beyond the Graph: Understanding Functionality

It’s crucial to remember that the vertical line test is a diagnostic tool, not a definitive proof of function status. It reveals whether a graph fails the test, but it doesn’t inherently guarantee that a relation isn’t a function. The test hinges on the visual representation of the relation, and as we’ve seen, certain representations – like implicit curves or parametric equations – can produce graphs that appear to fail the test despite being fundamentally functions. The key lies in recognizing that a function must possess a unique output for every input, and the vertical line test is a convenient way to visually assess this property within the context of a plotted graph.

Furthermore, the test’s effectiveness is directly tied to the domain and range of the relation. A relation might be a function within a specific domain, but fail the test when considered over a broader range. Consider, for instance, a function defined by (y = \sqrt{x}) for (x \ge 0). This is a perfectly valid function. However, if we extend the domain to include negative values of x, the function becomes undefined, and the vertical line test would correctly identify it as not a function.

Conclusion

The vertical line test remains a valuable and intuitive tool for determining whether a graph represents a function. However, its limitations must be acknowledged. It’s most effective for relations expressed as subsets of the Cartesian plane and relies on visual inspection. Understanding the underlying principles – that a function requires a single output for each input – and recognizing scenarios where the test might fail (such as implicit curves and parametric representations) are essential for a complete grasp of function concepts. By combining the test with a deeper understanding of the relation’s definition and domain, students can confidently determine whether a given graph truly embodies the characteristics of a function.