Math Problem Statement
what type of function is y=1/x^2+1
Solution
The function is a rational function, because it is the ratio of two polynomials. Specifically, it's the ratio of the constant over the quadratic expression .
Characteristics of this function:
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Domain: The function is defined for all real numbers , because the denominator is always positive and never equals zero (there are no real roots for ).
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Range: The output (or range) of the function is bounded between . The value of is always positive, but never zero, and the largest value occurs at , where .
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Asymptotes:
- Horizontal asymptote: As or , the value of the function approaches , so there is a horizontal asymptote at .
- Vertical asymptote: There are no vertical asymptotes, since the denominator never equals zero.
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Even function: The function is symmetric about the y-axis because . This means it is an even function.
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Decreasing behavior: The function is strictly decreasing for and for , meaning it becomes smaller as increases.
Would you like further details on how to derive any of these characteristics or ask any specific questions about the function?
Here are five related questions:
- What are the properties of rational functions in general?
- How do you determine the horizontal asymptote of a rational function?
- Can you find the first derivative of to determine the rate of change?
- How do we graph rational functions with asymptotes?
- How do you solve rational inequalities involving similar functions?
Tip: When analyzing rational functions, always check the domain first by ensuring the denominator does not equal zero!
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Math Problem Analysis
Mathematical Concepts
Rational Functions
Algebra
Asymptotes
Symmetry
Domain and Range
Formulas
y = 1 / (x^2 + 1)
Theorems
Properties of Rational Functions
Horizontal Asymptotes
Even Functions
Suitable Grade Level
Grades 10-12