Math Problem Statement
F(x)=5x +2 over x-7
Solution
The function you provided is:
This is a rational function, where the numerator is and the denominator is .
Key Properties:
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Domain: The function is undefined when the denominator is zero. So, implies . Therefore, the domain of is all real numbers except .
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Vertical Asymptote: The vertical asymptote occurs where the denominator is zero (which is the same as where the function is undefined). Therefore, there is a vertical asymptote at .
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Horizontal Asymptote: To find the horizontal asymptote, we compare the degrees of the polynomials in the numerator and the denominator.
- The degree of the numerator is 1.
- The degree of the denominator is also 1.
Since the degrees are equal, the horizontal asymptote is determined by the ratio of the leading coefficients: Therefore, the horizontal asymptote is .
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Intercepts:
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Y-intercept: Set to find the y-intercept. So, the y-intercept is .
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X-intercept: Set the numerator equal to zero to find the x-intercept. So, the x-intercept is .
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Would you like a more detailed explanation of any part of this function?
Here are some related questions you might find helpful:
- How do you determine slant asymptotes in rational functions?
- What is the behavior of the function as approaches the vertical asymptote?
- How can you graph this rational function?
- What are the steps to find the inverse of this function?
- Can you explain how to factor the numerator and denominator in more complex rational functions?
Tip: Always check the domain of a function before performing operations like addition, subtraction, or composition with other functions.
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Math Problem Analysis
Mathematical Concepts
Rational functions
Domain and range
Asymptotes
Intercepts
Formulas
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Theorems
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Suitable Grade Level
High school
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