Math Problem Statement

Based on the graph of f(x), determine where it is continuous.

Solution

The graph provided shows f(x)f(x), and we are tasked with determining where f(x)f(x) is continuous. A function is continuous if there are no breaks, jumps, or holes in the graph.

Observations from the Graph:

  1. There are visible discontinuities:

    • At x=6x = -6: A jump discontinuity.
    • At x=3x = 3: A hole (removable discontinuity).
  2. The function appears continuous elsewhere.

Continuity:

The function f(x)f(x) is continuous:

  • On the intervals (,6)(-\infty, -6), (6,3)(-6, 3), and (3,)(3, \infty).

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Related Questions:

  1. What are the different types of discontinuities in a graph?
  2. How do you test for continuity algebraically at a specific point?
  3. Why is a jump discontinuity different from a removable discontinuity?
  4. How can you modify f(x)f(x) to make it continuous everywhere?
  5. What real-world problems rely on continuous functions?

Tip: Always examine both the left-hand and right-hand limits to confirm continuity at a point.

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Math Problem Analysis

Mathematical Concepts

Graph Analysis
Continuity of Functions
Types of Discontinuities

Formulas

N/A (graph-based analysis)

Theorems

Definition of Continuity
Intermediate Value Theorem

Suitable Grade Level

Grades 11-12