Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

3. Functions

Intro to Functions & Their Graphs

College Algebra

3. Functions

Intro to Functions & Their Graphs

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2:21 minutes

Problem 16

Textbook Question

Determine the intervals of the domain over which each function is continuous. See Example 1.

Verified step by step guidance

Identify the type of function given. For example, is it a polynomial, rational, exponential, etc.?

Recall the general rules for continuity: Polynomials are continuous everywhere, rational functions are continuous everywhere except where the denominator is zero, etc.

If the function is rational, set the denominator equal to zero and solve for the values of x that make the denominator zero.

Determine the intervals of x where the function is not continuous based on the solutions from the previous step.

Express the intervals of continuity in interval notation, excluding any points where the function is not continuous.

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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.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. Understanding the domain is essential for analyzing continuity, as discontinuities often arise from values that are not included in the domain, such as division by zero or taking the square root of a negative number.

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 redefining it. Jump discontinuities occur when there is a sudden change in function value, while infinite discontinuities happen when the function approaches infinity at a certain point. Identifying these types helps in determining the intervals of continuity.