Math Problem Statement
Solution
The image contains two rational functions for which we need to determine the domain, range, and asymptotes.
Function 1:
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Domain: The domain of a rational function is all real numbers except where the denominator is zero. So, the domain is .
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Range: To find the range, we can use the fact that horizontal asymptotes can give insight into the behavior of the function as . Since the degrees of the numerator and denominator are the same, the horizontal asymptote is determined by the ratio of the leading coefficients: However, since the function never actually reaches the horizontal asymptote but can approach any value above or below 1, the range is .
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Asymptotes:
- Vertical Asymptote: Occurs where the denominator is zero, which is .
- Horizontal Asymptote: .
Function 2:
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Domain: We need to find where the denominator equals zero to exclude these values from the domain. Solving this quadratic equation using the quadratic formula: So, the domain is .
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Range: Finding the range analytically for more complex rational functions is challenging, but we can use a numerical or graphical approach. Generally, it involves investigating the behavior of the function as .
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Asymptotes:
- Vertical Asymptotes: These occur where the denominator is zero, which gives vertical asymptotes at and .
- Horizontal Asymptote: Since the degrees of the numerator and denominator are equal, the horizontal asymptote is determined by the ratio of the leading coefficients:
Summary:
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For :
- Domain:
- Range:
- Vertical Asymptote:
- Horizontal Asymptote:
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For :
- Domain:
- Range: Needs further analysis or graphical investigation.
- Vertical Asymptotes: and
- Horizontal Asymptote:
Let me know if you'd like further details or a graphical analysis!
Here are five related questions to expand your understanding:
- How do we determine slant (oblique) asymptotes for rational functions?
- What methods can be used to find the range of a rational function besides graphical approaches?
- How do horizontal asymptotes affect the long-term behavior of a rational function?
- What are removable discontinuities in rational functions, and how can they be identified?
- How does the degree of the numerator and denominator influence the behavior of a rational function?
Tip: Always check for common factors in the numerator and denominator before determining vertical asymptotes. Sometimes a factor can cancel, resulting in a removable discontinuity instead of an asymptote.
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Math Problem Analysis
Mathematical Concepts
Rational Functions
Asymptotes
Domain and Range
Formulas
Domain: Set where the denominator is not equal to zero.
Range: Horizontal asymptotes and function behavior as x approaches infinity.
Vertical Asymptotes: Occur where the denominator is zero and the function is undefined.
Horizontal Asymptote: Ratio of leading coefficients for same-degree polynomials.
Theorems
Quadratic Formula
Asymptote Properties
Suitable Grade Level
Grade 11-12
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