Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

0. Review of College Algebra

Functions

Trigonometry

0. Review of College Algebra

Functions

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2:16 minutes

Problem 81

Textbook Question

Determine the intervals of the domain over which each function is continuous. See Example 9.

Verified step by step guidance

Step 1: Identify the function type. Determine if the function is a polynomial, rational, trigonometric, exponential, or logarithmic function.

Step 2: Recall the continuity rules for each function type. For example, polynomial functions are continuous everywhere, while rational functions are continuous everywhere except where the denominator is zero.

Step 3: If the function is rational, set the denominator equal to zero and solve for the variable to find points of discontinuity.

Step 4: For trigonometric functions, identify any vertical asymptotes or points where the function is undefined, such as where $ \tan(x) $ or $ \sec(x) $ have vertical asymptotes.

Step 5: Combine the information from the previous steps to determine the intervals of continuity, excluding any points of discontinuity identified.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval. This concept is crucial for determining where a function does not have breaks, jumps, or asymptotes.

Types of Discontinuities

Discontinuities can be classified into three main types: removable, jump, and infinite. A removable discontinuity occurs when a function is not defined at a point but can be made continuous by defining it appropriately. Jump discontinuities happen when the left-hand and right-hand limits at a point do not match, while infinite discontinuities occur when a function approaches infinity at a certain point.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses brackets [ ] to include endpoints and parentheses ( ) to exclude them. Understanding interval notation is essential for expressing the domain of continuity for functions, as it succinctly conveys which intervals are valid for the function's continuity.