Math Problem Statement
Find the range of 3x−2
x+5
Solution
To find the range of the function , we need to determine all possible values that the function can take for different values of .
Step 1: Identify Domain Restrictions
First, the function is undefined when the denominator is zero, i.e., when:
Thus, the function is undefined at , and the domain is .
Step 2: Analyze the Horizontal Asymptote
To find the horizontal asymptote, consider the behavior of the function as or . Divide both the numerator and denominator by :
As or , the terms and approach zero, so:
Thus, the function has a horizontal asymptote at .
Step 3: Check for Critical Points
To find the critical points, set and solve for in terms of :
Multiply both sides by :
Expand both sides:
Now, isolate :
Factor out :
Solve for :
Step 4: Exclude Values for Which
The function is undefined at . Set the denominator equal to zero to find which -value is not possible:
Thus, is not in the range, since as , but it never equals 3.
Conclusion: The Range
The range of the function is all real numbers except :
Would you like more details on this process?
Here are 5 related questions to explore further:
- What is the domain of the function ?
- How do you find the vertical asymptote of a rational function?
- How do you graph a rational function and identify its asymptotes?
- What is the general method to find the range of any rational function?
- How does the range change if the numerator and denominator have higher degrees?
Tip: Always check for asymptotes (vertical and horizontal) when analyzing rational functions, as they provide insight into the function's range and behavior.
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Math Problem Analysis
Mathematical Concepts
Algebra
Rational Functions
Asymptotes
Formulas
f(x) = (3x - 2) / (x + 5)
Horizontal Asymptote: y = 3
x = (-5y - 2) / (y - 3)
Theorems
Asymptote Theorem: Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity.
Suitable Grade Level
Grades 10-12