Math Problem Statement
(x-2)²/(4x²+16x) find the range
Solution
To find the range of the function we can follow these steps:
Step 1: Simplify the function
We first simplify the denominator:
So the function becomes:
Step 2: Analyze the domain
The denominator cannot be zero, so we exclude the values and from the domain. Thus, the domain is all real numbers except and .
Step 3: Determine behavior at the critical points
Now, let's analyze the behavior of for large values of , and near the undefined points and .
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As or : For large values of , the highest power terms dominate, and we get:
So, approaches as or .
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At and : The function is undefined at these points.
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As from either side: The denominator approaches zero and the numerator is finite, so or .
Step 4: Check for minimum values
Since , the numerator is always non-negative. Thus, .
At , the numerator is zero, so .
Step 5: Conclusion about the range
The function approaches as the minimum value at and approaches for large . The function also becomes unbounded near . Hence, the range of the function is:
Would you like a more detailed breakdown or have any questions?
Here are some related questions you might find helpful:
- How do you find the domain of a rational function?
- What happens to a function at vertical asymptotes?
- How can you determine if a function has a horizontal asymptote?
- How do you find the critical points of a function?
- 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
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