Introduction to Relations and Functions: Videos & Practice Problems

Relations connect inputs (x-values) to outputs (y-values) and can be represented as ordered pairs, tables, graphs, or equations. A function is a special relation where each input corresponds to exactly one output. The vertical line test helps identify functions on graphs: if any vertical line crosses the graph more than once, it is not a function. The domain is the set of all input values, and the range is the set of all output values. For continuous graphs, domain and range are expressed using interval notation, indicating the span of x and y values covered.

Intro to Relations and Functions

A relation is a fundamental concept in mathematics that describes a connection between inputs (commonly represented as x values) and outputs (represented as y values). These connections are often expressed as a set of ordered pairs in the form \((x, y)\), where each pair shows how an input corresponds to an output. For example, if you have the ordered pairs \((-2, 2)\), \((1, 1)\), \((3, -2)\), and \((1, 4)\), the inputs are the unique x values \(-2\), \$1\(, and \)3\(, while the outputs are the y values \)2\(, \)1\(, \)-2\(, and \)4\(. Visualizing these relations can be done by drawing arrows from each input to its corresponding output, illustrating the mapping clearly.

A function is a special type of relation where each input is paired with exactly one output. This means no input value maps to more than one output value. For instance, consider the ordered pairs \)(-4, 2)\(, \)(-2, -1)\(, \)(1, 2)\(, and \)(3, 4)\(. Here, each input \)-4\(, \)-2\(, \)1\(, and \)3\( corresponds to a single output, making this relation a function. In contrast, if an input like \)1\( maps to both \)1\( and \)4$, as in the previous example, the relation is not a function because the input has multiple outputs.

Understanding the distinction between relations and functions is crucial: all functions are relations, but not all relations qualify as functions. Relations and functions can be represented in various ways, including sets of ordered pairs, tables, graphs, or equations. This flexibility allows for diverse approaches to analyzing and interpreting these mathematical concepts.

In summary, recognizing whether a relation is a function involves checking if each input has a unique output. This foundational idea supports further exploration of mathematical modeling, graphing, and problem-solving involving functions and relations.

Intro to Relations and Functions Example 1

To determine whether a relation is a function, it is essential to analyze the mapping between inputs (often represented as x-values) and outputs (y-values). A function is defined as a relation where each input corresponds to exactly one output. This means no single input can map to multiple outputs.

For example, when given a table of values, start by listing the unique inputs. Then, observe the outputs associated with each input. If any input is paired with more than one output, the relation is not a function. For instance, if the input ½ maps to both 4 and 5, this violates the definition of a function because one input has multiple outputs.

Conversely, if every input maps to only one output, even if the outputs are represented by variables or letters rather than specific numbers, the relation qualifies as a function. The key is the uniqueness of the output for each input, not the specific values of the outputs.

In summary, to verify if a relation is a function, check whether any input value corresponds to more than one output value. If it does, the relation is not a function; if it does not, the relation is a function. This fundamental concept is crucial in understanding functions in algebra and their applications.

Vertical Line Test

A function is a special type of relation where each input corresponds to exactly one unique output. To quickly determine whether a relation represented on a graph is a function, the vertical line test is a useful and efficient method. This test applies only when the relation is graphed, and it involves imagining or drawing vertical lines across the graph.

The vertical line test states that if any vertical line intersects the graph at more than one point, the relation is not a function. This is because multiple intersections indicate that a single input (x-value) maps to multiple outputs (y-values), violating the definition of a function. Conversely, if every vertical line touches the graph at most once, the relation qualifies as a function.

For example, when examining a graph with discrete points, drawing vertical lines through each point helps verify if any line crosses more than one point. If no vertical line intersects more than one point, the graph passes the vertical line test and represents a function. In contrast, for graphs depicting smooth curves with infinitely many points, the test remains effective. If a vertical line intersects the curve at two or more points, it confirms the relation is not a function.

Understanding the vertical line test enhances the ability to analyze graphs quickly and accurately, reinforcing the fundamental concept that functions assign exactly one output to each input. This test is a practical tool in algebra and precalculus for distinguishing functions from general relations.

State the inputs and outputs of the following relation. Is it a function?
$\{(-3,5),(0,2),(3,5)\}$

• A.
• B.
• C.
• D.

Answer: (C)

State the inputs and outputs of the following relation. Is it a function?
$\{(2,5),(0,2),(2,9)\}$

• A.
• B.
• C.
• D.

Answer: (B)

Vertical Line Test Example 2

To determine whether a graph represents a function, the vertical line test is a useful method. This test states that if a vertical line can be drawn anywhere on the graph and it intersects the graph at more than one point, then the graph is not a function. Conversely, if every vertical line drawn intersects the graph at most once, then the graph is indeed a function.

For example, consider three different graphs:

1. **Graph A**: When applying the vertical line test, any vertical line drawn across this graph intersects it at only one point. This consistent behavior confirms that Graph A is a function.

2. **Graph B**: In this case, a vertical line can be drawn that intersects the graph at multiple points. This indicates that Graph B does not satisfy the criteria for a function.

3. **Graph C**: Similar to Graph A, any vertical line drawn on Graph C intersects it at only one point. Thus, Graph C is also classified as a function.

In summary, Graphs A and C are functions, while Graph B is not. This understanding of the vertical line test is essential for identifying functions in various mathematical contexts.

Domain and Range

Understanding the domain and range is essential when working with relations and functions. The domain refers to the set of all possible input values, typically represented as the x-values, while the range is the set of all possible output values, or the y-values. These concepts help describe the behavior and limitations of functions and relations.

When given a relation as a set of ordered pairs or a table, finding the domain and range involves listing all the distinct x-values for the domain and all the distinct y-values for the range. For example, if a relation consists of the ordered pairs \((-2, 2)\), \((1, 1)\), \((3, -2)\), and \((4, 4)\), the domain is \(\{-2, 1, 3, 4\}\) and the range is \(\{2, 1, -2, 4\}\). This straightforward approach works well for discrete data points.

However, when dealing with continuous functions represented by graphs, the domain and range often include infinitely many values. In such cases, it is more practical to express the domain and range using interval notation. Interval notation concisely describes all values between two endpoints, using brackets [ ] to include endpoints and parentheses ( ) to exclude them. For instance, if a graph starts at \(x = -4\) (included) and extends indefinitely to the right, the domain is written as \([-4, \infty)\). Similarly, if the range starts at \(y = -2\) (included) and extends upward without bound, the range is \([-2, \infty)\).

It is important to recognize that infinity (\(\infty\)) is never included in the interval, so it is always paired with a parenthesis. Solid dots on a graph indicate that the endpoint is included in the domain or range, while open dots mean the endpoint is excluded.

In summary, the domain encompasses all input values (x-values) for which the function or relation is defined, and the range includes all output values (y-values) that result from those inputs. Whether working with discrete sets of ordered pairs or continuous graphs, identifying the domain and range is fundamental to understanding the scope and behavior of functions in mathematics.

Find the domain and range for each relation.
(A) $\{(-4,2),(-3,-5),(3,4),(5,-2)\}$

• A. Domain: $\{-4,-3,3,5\}$, Range: $\{-4,-3,2,5\}$
• B. Domain: $\left\lbrace-4,-3,3,5\right\rbrace$, Range: $\{2,-5,4,-2\}$
• C. Domain: ,$\{-4,2,3,5\}$ Range: $\{2,-5,4,-2\}$
• D. Domain: $\left\lbrace-4,-3,3,5\right\rbrace$, Range: $\{5,-2,4,-5\}$

Answer: (B)

Find the domain and range for each relation.
(B)

• A. Domain: $\left\lbrace-4,2,3\right\rbrace$, Range: $\left\lbrace2,3,-4\right\rbrace$
• B. Domain: $\left\lbrace-4,-3,2,3\right\rbrace$, Range: $\left\lbrace1,2,-3,3,-4\right\rbrace$$\left\lbrace2,-3,3,-4\right\rbrace$
• C. Domain: $\left\lbrace-4,-3,3\right\rbrace$, Range: $\left\lbrace2,-3,3,-4\right\rbrace$
• D. Domain: $\{-4,-3,2,3\}$, Range: $\{2,-3,3,-4\}$

Answer: (D)

Domain and Range Example 3

To determine the domain and range of a function from its graph, it is essential to understand that the domain represents all possible x-values for which the function is defined, while the range represents all possible y-values that the function attains. Expressing these sets in interval notation provides a clear and concise way to describe them.

For example, consider a graph where the function exists starting just to the right of x = -4 and continues up to and including x = 2. Since there is an open circle at x = -4, this value is not included in the domain, whereas a closed circle at x = 2 means this endpoint is included. Therefore, the domain in interval notation is \((-4, 2]\).

Examining the range of this function, suppose the y-values start just above y = -4 (indicated by an open circle, so not included) and extend up to and include y = 5 (indicated by a closed circle). This means the range is \((-4, 5]\). The open and closed circles are crucial for determining whether endpoints are included or excluded in the intervals.

In contrast, consider a graph that extends infinitely in all directions, both horizontally and vertically, as indicated by arrows at the ends of the curve. This implies that the function is defined for all real numbers along the x-axis and attains all real values along the y-axis. Hence, the domain and range are both all real numbers, expressed as \((-\infty, \infty)\). There are no restrictions or boundaries limiting the function's input or output values.

Understanding how to interpret open and closed circles on a graph, as well as recognizing infinite extensions, allows for accurate identification of domain and range. This skill is fundamental in analyzing functions and their behaviors, providing a foundation for further study in calculus and algebra.

Here’s what students ask on this topic:

What is the difference between a relation and a function in algebra?

A relation in algebra is a set of ordered pairs that connects inputs (x-values) to outputs (y-values). It can be represented as ordered pairs, tables, graphs, or equations. A function is a special type of relation where each input corresponds to exactly one output. This means that in a function, no input value is paired with more than one output value. For example, if an input x = 1 is paired with y = 2 and y = 3, this relation is not a function. However, if every input has only one output, then it is a function. All functions are relations, but not all relations are functions. Understanding this distinction is fundamental in algebra and helps in analyzing mathematical models and graphs.

How do you use the vertical line test to determine if a graph represents a function?

The vertical line test is a quick method to determine if a graph represents a function. To use it, imagine drawing vertical lines (lines parallel to the y-axis) across the graph. If any vertical line crosses the graph at more than one point, the graph does not represent a function. This is because multiple points on the same vertical line mean one input (x-value) corresponds to multiple outputs (y-values), violating the definition of a function. If every vertical line crosses the graph at most once, then the graph represents a function. This test is especially useful for visualizing functions and relations on coordinate planes.

What are domain and range in the context of relations and functions?

In relations and functions, the domain is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values). For example, if a function is given as a set of ordered pairs, the domain consists of all the first elements of these pairs, and the range consists of all the second elements. When dealing with continuous graphs, the domain and range are often expressed using interval notation to represent all x and y values covered by the graph. Understanding domain and range helps in analyzing the behavior and limitations of functions.

How do you find the domain and range from a graph using interval notation?

To find the domain and range from a graph using interval notation, first identify the span of x-values (domain) and y-values (range) covered by the graph. For the domain, look at the horizontal extent of the graph: find the smallest and largest x-values where the graph exists. Use brackets $[$ or $]$ to include endpoints if the graph touches those points (solid dots), and parentheses $($ or $)$ to exclude endpoints if the graph does not include them (open dots). For the range, do the same by looking at the vertical extent of the graph. For example, if the graph starts at x = -4 (included) and extends to infinity, the domain is $[-4,\infty )$. This method efficiently describes all input and output values for continuous functions.

Can a relation have repeated input values, and how does that affect whether it is a function?

Yes, a relation can have repeated input values, but whether it is a function depends on how those inputs relate to outputs. If a repeated input value corresponds to the same output each time, the relation can still be a function. However, if a repeated input value corresponds to different outputs, then the relation is not a function. For example, if x = 1 maps to y = 2 and also to y = 4, this violates the function rule that each input has exactly one output. Therefore, repeated inputs are allowed in relations, but for functions, each input must have a unique output.