Introduction to Functions: Videos & Practice Problems

Welcome to the world of business calculus, where you'll explore essential mathematical concepts that are crucial for understanding and applying calculus in a business context. This course will guide you through key topics such as derivatives, integrals, related rates, optimization, and the fundamentals of differential equations.

Derivatives are a fundamental concept in calculus, representing the rate of change of a function. They are essential for understanding how quantities change in relation to one another, which is particularly useful in business for analyzing trends and making informed decisions. The derivative of a function \( f(x) \) is denoted as \( f'(x) \) or \( \frac{df}{dx} \), and it can be calculated using various rules such as the power rule, product rule, and quotient rule.

Integrals, on the other hand, are used to find the total accumulation of a quantity, which can be applied to determine areas under curves or total revenue over time. The integral of a function \( f(x) \) is represented as \( \int f(x) \, dx \), and understanding definite and indefinite integrals is crucial for solving real-world problems in business scenarios.

In addition to these core concepts, the course will cover applications such as related rates, which involve finding the rate at which one quantity changes in relation to another, and optimization, which focuses on finding maximum or minimum values of functions to make the best business decisions. You'll also be introduced to differential equations, which model relationships involving rates of change and are widely used in various fields, including economics and finance.

Throughout the course, you'll engage with concept videos that break down complex ideas into manageable parts, along with example problems that reinforce your understanding. Practice problems will allow you to apply what you've learned, and video solutions will guide you through the problem-solving process. Additionally, downloadable worksheets will help you take notes and review key concepts at your own pace.

As you embark on this journey through business calculus, remember that support is available whenever you need it. By actively participating and practicing, you'll build the skills and confidence necessary to excel in your course and beyond. Let's dive into the fascinating world of business calculus together!

Understanding the concepts of relations and functions is essential 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 crucial: 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 (X value) is associated with multiple outputs (Y values), 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 appears more than once with different Y values, the relation fails the function test.

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: the first graph fails the vertical line test, indicating that it is not a function, while the second graph passes the test, confirming it as a function. This method simplifies the process of determining the nature of relations and functions without the need for exhaustive listing of inputs and outputs.

In summary, recognizing the difference between relations and functions, and applying the vertical line test, are key skills in understanding mathematical relationships and graphing. These concepts form the foundation for more advanced studies in mathematics, including algebra and calculus.

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, the graph is a function.

For the first graph (graph A), applying the vertical line test shows that no matter where a vertical line is drawn, it only intersects the graph at a single point. Therefore, graph A qualifies as a function.

In contrast, when examining graph B, a vertical line can be drawn that intersects the graph at multiple points. This indicates that graph B does not meet the criteria for a function.

Finally, for graph C, similar to graph A, every vertical line drawn intersects the graph at only one point. Thus, graph C is also classified as a function.

In summary, the graphs that are functions are graph A and graph C.

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 graph covers. 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 graph extends from -4 to 0 and from 1 to 4, it can be expressed as:

Domain: \([-4, 0) \cup (1, 4]\

Here, the union symbol (∪) connects multiple intervals. The range can be similarly expressed. If the range spans from -3 to -1 and from 1 to 3, it would be written as:

Range: \([-3, -1) \cup [1, 3]\

In this case, -3 is included, while -1 is not, and both 1 and 3 are included. This notation is crucial for accurately conveying the values that are part of the domain and range.

In summary, to find the domain and range from a graph, apply the squish strategy to visualize the x and y values, and use interval notation to express your findings clearly. This method not only simplifies the process but also enhances your understanding of the function's characteristics.