Math Problem Statement

f(x)=8x/x^2+4

Solution

The given function is:

f(x)=8xx2+4f(x) = \frac{8x}{x^2 + 4}

Simplifying the Function

At first glance, the function is already in its simplest form. The numerator is 8x8x, and the denominator is x2+4x^2 + 4. 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 xx that would make the denominator zero. In this case, x2+4=0x^2 + 4 = 0 leads to:

x2=4x^2 = -4

Since there are no real numbers whose square is negative, the denominator is never zero for any real xx. Therefore, the domain of f(x)f(x) is all real numbers:

Domain: (,)\text{Domain: } (-\infty, \infty)

Asymptotes

  • Vertical Asymptotes: Since the denominator x2+4x^2 + 4 is never zero, there are no vertical asymptotes.
  • Horizontal Asymptote: As xx \to \infty or xx \to -\infty, the function behaves like 8xx2=8x\frac{8x}{x^2} = \frac{8}{x}, which approaches 0. Therefore, the horizontal asymptote is y=0y = 0.

Behavior of the Function

  • At x=0x = 0:

f(0)=8(0)02+4=0f(0) = \frac{8(0)}{0^2 + 4} = 0

  • As xx \to \infty or xx \to -\infty: As mentioned, the function tends towards 0, suggesting a horizontal asymptote at y=0y = 0.

Would you like further details or any specific clarification?

Related Questions:

  1. What are the critical points and extrema of the function f(x)f(x)?
  2. How do you find the derivative f(x)f'(x) of the function?
  3. Can you find the concavity of f(x)f(x) by calculating the second derivative f(x)f''(x)?
  4. What is the range of the function f(x)f(x)?
  5. How do you sketch the graph of f(x)f(x) 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