Math Problem Statement

Draw four graphs (A, B, C, and D) where the functions are defined for all x-values, and Graphs A, B, and C have the given properties.

Solution

It looks like you've uploaded an image of a mathematical problem involving the drawing of four graphs (A, B, C, and D). The task specifies that graphs A, B, and C must meet certain properties, and graph D should combine all these properties. Let's break down the properties and how they relate to each graph:

  1. Graph A:

    • The function must satisfy f(2)=2f(-2) = 2.
    • The limit as x2x \to -2 of f(x)f(x) must be 1-1.
    • The line x=2x = -2 is a vertical asymptote.
  2. Graph B:

    • The function must satisfy f(0)=1f(0) = 1.
    • The function is continuous at x=0x = 0.
  3. Graph C:

    • The function must satisfy f(3)=4f(3) = 4.
    • The limit as x3+x \to 3^+ of f(x)f(x) must be 44.
    • The limit as x3x \to 3^- of f(x)f(x) must be 22.

For graph D, you'll need to draw a function that satisfies all properties listed for A, B, and C.

To help you better visualize how these properties can be combined, I can walk through each graph's behavior in detail, and explain how to approach drawing them. Would you like me to go into this breakdown, or do you need additional guidance on any specific graph?

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Math Problem Analysis

Mathematical Concepts

Limits and Continuity
Piecewise Functions
Vertical Asymptotes

Formulas

lim x→c f(x) = L (Definition of a limit)
Continuity: f(x) is continuous at x=c if lim x→c f(x) = f(c)
Vertical Asymptote: A function has a vertical asymptote at x=a if lim x→a f(x) → ±∞

Theorems

Limit Theorems
Continuity Theorem
Definition of Asymptotes

Suitable Grade Level

Grades 10-12