Math Problem Statement
Solution
Let's analyze the graph and conditions provided to determine if it matches the description in the question:
Given Conditions:
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Domain:
- This means the function is not defined at and .
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Y-intercept: The function must pass through the point .
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Increasing on : The function is increasing on the interval .
Analysis of the Graph:
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Domain:
- From the graph, the function has vertical asymptotes at and , which matches the condition that the function is not defined at these points. Thus, the domain is indeed .
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Y-intercept:
- The graph clearly passes through the point , satisfying the y-intercept condition.
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Increasing on :
- In the interval , the graph shows that the function is indeed increasing, as the curve rises from at to a higher value near .
Conclusion:
The graph satisfies all the given conditions: it has the correct domain, the y-intercept at , and is increasing on the interval .
Would you like further clarification or details on any specific part?
Here are 5 follow-up questions to deepen your understanding:
- How can we express the range of this function based on the graph?
- What is the behavior of the function as and ?
- Can we deduce any symmetry in the graph?
- How does the graph behave as or ?
- 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
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Theorems
Domain Theorem for Rational Functions
Monotonicity
Suitable Grade Level
Grades 10-12
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