Math Problem Statement

(x-2)²/(4x²+16x) find the range

Solution

To find the range of the function f(x)=(x2)24x2+16x,f(x) = \frac{(x - 2)^2}{4x^2 + 16x}, we can follow these steps:

Step 1: Simplify the function

We first simplify the denominator:

4x2+16x=4x(x+4).4x^2 + 16x = 4x(x + 4).

So the function becomes:

f(x)=(x2)24x(x+4).f(x) = \frac{(x - 2)^2}{4x(x + 4)}.

Step 2: Analyze the domain

The denominator 4x(x+4)4x(x + 4) cannot be zero, so we exclude the values x=0x = 0 and x=4x = -4 from the domain. Thus, the domain is all real numbers except x=0x = 0 and x=4x = -4.

Step 3: Determine behavior at the critical points

Now, let's analyze the behavior of f(x)f(x) for large values of xx, and near the undefined points x=0x = 0 and x=4x = -4.

  1. As xx \to \infty or xx \to -\infty: For large values of xx, the highest power terms dominate, and we get:

    f(x)x24x2=14.f(x) \approx \frac{x^2}{4x^2} = \frac{1}{4}.

    So, f(x)f(x) approaches 14\frac{1}{4} as xx \to \infty or xx \to -\infty.

  2. At x=0x = 0 and x=4x = -4: The function is undefined at these points.

  3. As x4x \to -4 from either side: The denominator approaches zero and the numerator is finite, so f(x)f(x) \to \infty or f(x)f(x) \to -\infty.

Step 4: Check for minimum values

Since (x2)20(x - 2)^2 \geq 0, the numerator is always non-negative. Thus, f(x)0f(x) \geq 0.

At x=2x = 2, the numerator is zero, so f(2)=0f(2) = 0.

Step 5: Conclusion about the range

The function approaches 00 as the minimum value at x=2x = 2 and approaches 14\frac{1}{4} for large xx. The function also becomes unbounded near x=4x = -4. Hence, the range of the function is:

[0,).\boxed{[0, \infty)}.

Would you like a more detailed breakdown or have any questions?

Here are some related questions you might find helpful:

  1. How do you find the domain of a rational function?
  2. What happens to a function at vertical asymptotes?
  3. How can you determine if a function has a horizontal asymptote?
  4. How do you find the critical points of a function?
  5. What methods can be used to check the limits of a function?

Tip: Always check for vertical and horizontal asymptotes when working with rational functions.

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Range of a Function
Asymptotes

Formulas

f(x) = (x - 2)^2 / (4x^2 + 16x)
4x^2 + 16x = 4x(x + 4)

Theorems

Behavior of Rational Functions at Asymptotes
Limit of a Function as x approaches infinity
Critical Points and Behavior Analysis

Suitable Grade Level

Grades 10-12