Math Problem Statement
f(x)=8x/x^2+4
Solution
The given function is:
Simplifying the Function
At first glance, the function is already in its simplest form. The numerator is , and the denominator is . This is a rational function, and you can't simplify it any further.
Domain of the Function
To find the domain of the function, we look for values of that would make the denominator zero. In this case, leads to:
Since there are no real numbers whose square is negative, the denominator is never zero for any real . Therefore, the domain of is all real numbers:
Asymptotes
- Vertical Asymptotes: Since the denominator is never zero, there are no vertical asymptotes.
- Horizontal Asymptote: As or , the function behaves like , which approaches 0. Therefore, the horizontal asymptote is .
Behavior of the Function
- At :
- As or : As mentioned, the function tends towards 0, suggesting a horizontal asymptote at .
Would you like further details or any specific clarification?
Related Questions:
- What are the critical points and extrema of the function ?
- How do you find the derivative of the function?
- Can you find the concavity of by calculating the second derivative ?
- What is the range of the function ?
- How do you sketch the graph of with all key points?
Tip: When analyzing a rational function, always check for vertical and horizontal asymptotes to understand its long-term behavior.
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Math Problem Analysis
Mathematical Concepts
Rational Functions
Domain and Range
Asymptotes
Formulas
f(x) = 8x / (x^2 + 4)
Horizontal Asymptote: y = 0
Theorems
Asymptote Theorem
Rational Function Behavior
Suitable Grade Level
Grades 10-12
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