Math Problem Statement

Which graph represents a function that has the domain (-∞, -4) ∪ (-4, 3) ∪ (3, ∞), has a y-intercept at (0, 3), and is increasing on the interval (0, 4)?

Solution

Let's analyze the graph and conditions provided to determine if it matches the description in the question:

Given Conditions:

  1. Domain: (,4)(4,3)(3,)(-\infty, -4) \cup (-4, 3) \cup (3, \infty)

    • This means the function is not defined at x=4x = -4 and x=3x = 3.
  2. Y-intercept: The function must pass through the point (0,3)(0, 3).

  3. Increasing on (0,4)(0, 4): The function is increasing on the interval (0,4)(0, 4).

Analysis of the Graph:

  1. Domain:

    • From the graph, the function has vertical asymptotes at x=4x = -4 and x=3x = 3, which matches the condition that the function is not defined at these points. Thus, the domain is indeed (,4)(4,3)(3,)(-\infty, -4) \cup (-4, 3) \cup (3, \infty).
  2. Y-intercept:

    • The graph clearly passes through the point (0,3)(0, 3), satisfying the y-intercept condition.
  3. Increasing on (0,4)(0, 4):

    • In the interval (0,4)(0, 4), the graph shows that the function is indeed increasing, as the curve rises from y=3y = 3 at x=0x = 0 to a higher value near x=4x = 4.

Conclusion:

The graph satisfies all the given conditions: it has the correct domain, the y-intercept at (0,3)(0, 3), and is increasing on the interval (0,4)(0, 4).

Would you like further clarification or details on any specific part?


Here are 5 follow-up questions to deepen your understanding:

  1. How can we express the range of this function based on the graph?
  2. What is the behavior of the function as x4x \to -4 and x3x \to 3?
  3. Can we deduce any symmetry in the graph?
  4. How does the graph behave as xx \to \infty or xx \to -\infty?
  5. What would the derivative of this function tell us about its increasing and decreasing behavior?

Tip: Always look for vertical asymptotes to identify where a function is undefined, which helps in determining the domain.

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Math Problem Analysis

Mathematical Concepts

Domain and Range
Vertical Asymptotes
Intervals of Increase and Decrease
Intercepts

Formulas

-

Theorems

Domain Theorem for Rational Functions
Monotonicity

Suitable Grade Level

Grades 10-12