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 15

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 represented by the graph. In this case, it is a linear function.

Observe the graph to determine if there are any discontinuities. A discontinuity occurs where the function is not defined or has a break.

Notice the open circle at the point (1, 13). This indicates that the function is not defined at x = 1.

Determine the intervals where the function is continuous. Since the function is linear and only has a discontinuity at x = 1, it is continuous everywhere else.

Express the intervals of continuity. The function is continuous on the intervals (-∞, 1) and (1, ∞).

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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, meaning there are no breaks, jumps, or holes in the graph.

Domain of a Function

The domain of a function is the complete set of possible values (inputs) for which the function is defined. Understanding the domain is crucial for determining where a function is continuous, as discontinuities can occur at points not included in the domain.

Graphical Interpretation

Analyzing the graph of a function provides visual insight into its behavior, including continuity. Points where the graph is not connected or has holes indicate discontinuities, while a smooth, unbroken line suggests continuity across the interval. The graph in the question shows a point at (1, 13) that is open, indicating a discontinuity at that specific point.