Introduction to Functions: Videos & Practice Problems

Relations connect X and Y values as ordered pairs, while functions are specific relations where each input has at most one output. To determine if a relation is a function, use the vertical line test: if a vertical line intersects the graph at more than one point, it is not a function. Additionally, the domain represents allowed X values, and the range represents allowed Y values. The squish strategy helps visualize these by compressing the graph to the X or Y axis, respectively, and interval notation or set builder notation can express the domain and range clearly.

Introduction to Calculus Channel

Welcome to your journey in calculus! This course is designed to help you master essential concepts such as derivatives, integrals, and their applications, including related rates, optimization, and the fundamentals of differential equations. With a focus on understanding and applying these concepts, you'll be well-prepared for your exams.

Each chapter aligns with your specific class and textbook, featuring concept videos that delve into the necessary equations and problem-solving techniques. Real-world problems are integrated into the lessons to enhance your skills and build confidence. You will also have the opportunity to tackle practice problems, each accompanied by a video solution to guide you through the process of finding the correct answer.

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Relations and Functions

Understanding the concepts of relations and functions is crucial in mathematics, particularly when analyzing graphs. A relation is defined as a connection between sets of values, specifically X (input) and Y (output) values, which can be represented graphically as ordered pairs. In contrast, a function is a specific type of relation where each input corresponds to at most one output. This distinction is essential: while all functions are relations, not all relations qualify as functions.

To determine whether a relation is a function, one can analyze the inputs and outputs. For instance, if a particular input has multiple outputs, the relation cannot be classified as a function. This can be illustrated through examples where the X values are listed alongside their corresponding Y values. If any X value is linked to more than one Y value, the relation fails the function criteria.

A more efficient method to assess whether a relation is a function is through the vertical line test. This test states that if a vertical line can be drawn on a graph that intersects the graph at more than one point, the relation is not a function. Conversely, if every vertical line drawn intersects the graph at only one point, the relation is indeed a function.

For example, consider two graphs: if a vertical line drawn through any point on the first graph intersects at multiple points, it indicates that the relation is not a function. In contrast, if vertical lines drawn through the second graph only intersect at single points, it confirms that the relation is a function.

In summary, recognizing the difference between relations and functions, and applying the vertical line test, are key skills in understanding how to analyze and interpret graphs effectively. This foundational knowledge is vital for further studies in mathematics and its applications.

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

• A.
• B.
• C.
• D.

Answer: (C)

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

• A.
• B.
• C.
• D.

Answer: (B)

Relations & Functions Example 1

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 only one point, it qualifies as a function.

In analyzing the graphs:

For Graph A, when vertical lines are drawn at various locations, each line intersects the graph at only one point. This consistent result confirms that Graph A is indeed a function.

In contrast, Graph B fails the vertical line test. A vertical line drawn at certain positions intersects the graph at multiple points, indicating that it is not a function.

Finally, Graph C passes the vertical line test as well. Similar to Graph A, any vertical line drawn intersects the graph at only one point, confirming that Graph C is a function.

In summary, the graphs that qualify as functions are Graph A and Graph C.

Finding the Domain and Range of a Graph

Understanding the domain and range of a function is essential in analyzing its behavior through its graph. The domain refers to the set of all possible x-values, while the range encompasses all possible y-values. A helpful technique for determining these sets is the "squish strategy." To find the domain, visualize squishing the graph down to the x-axis, which reveals the x-values that the function can take. Conversely, squishing the graph down to the y-axis helps identify the range.

When expressing the domain and range, interval notation is commonly used. This notation employs brackets and parentheses: brackets [ ] indicate that a value is included (closed dot), while parentheses ( ) signify that a value is excluded (open circle). For example, if the domain of a function spans from -4 to 5, it can be expressed as [-4, 0) ∪ (1, 4], where the union symbol (∪) connects multiple intervals. Here, the interval from -4 to 0 includes both endpoints, while the interval from 1 to 4 includes 1 but excludes 0.

Set builder notation is another way to represent the domain and range, using inequality symbols. For instance, the domain can be expressed as {x | -4 ≤ x < 0 or 1 ≤ x ≤ 4}, indicating the conditions for x-values. It is crucial to note that closed circles or solid lines indicate included values, while open circles denote excluded values.

To illustrate, consider a graph where the domain is determined by squishing it down to the x-axis, revealing intervals such as [-4, 0) and (1, 4]. The range is found by squishing down to the y-axis, resulting in intervals like [-3, -1) and (1, 3]. This methodical approach allows for a clear understanding of the function's behavior and the values it can take.

Find the domain and range of the following graph (write your answer using interval notation).

• A. Dom: $\left[-5,5\right]$ , Ran: $\left\lbrack-4,4\right\rbrack$
• B. Dom: $\left[-2,2\right]$ , Ran: $\left[-3,3\right]$
• C. Dom: $\left[-4,4\right]$ , Ran: $\left[-5,5\right]$
• D. Dom: $\left(-5,5\right)$ , Ran: $\left(-4,4\right)$

Answer: (C)

Here’s what students ask on this topic:

What is the difference between a relation and a function?

A relation is a connection between X and Y values, represented as ordered pairs. A function is a specific type of relation where each input (X value) has at most one output (Y value). In other words, for a relation to be a function, no X value can be associated with more than one Y value. All functions are relations, but not all relations are functions.

How can you determine if a relation is a function using the vertical line test?

The vertical line test is a method to determine if a relation is a function. To apply this test, draw vertical lines through the graph of the relation. If any vertical line intersects the graph at more than one point, the relation is not a function. If every vertical line intersects the graph at most once, the relation is a function.

What is the domain and range of a function?

The domain of a function is the set of all possible input values (X values) that the function can accept. The range is the set of all possible output values (Y values) that the function can produce. To find the domain, look at the allowed X values on the graph. To find the range, look at the allowed Y values on the graph.

How do you find the domain and range of a function using the squish strategy?

The squish strategy helps visualize the domain and range of a function. To find the domain, squish the graph down to the X axis, which shows the allowed X values. To find the range, squish the graph down to the Y axis, which shows the allowed Y values. This method simplifies identifying the domain and range from the graph.

What is interval notation and how is it used to express the domain and range of a function?

Interval notation is a way to represent the domain and range of a function using brackets and parentheses. Brackets [ ] indicate that the endpoint is included, while parentheses ( ) indicate that the endpoint is not included. For example, if the domain of a function is all X values from -4 to 5, it is written as [-4, 5]. If the range is all Y values from -1 to 2, it is written as [-1, 2].