Math Problem Statement
Consider the function � ( � )
� + 3 � − 1 f(x)= x−1 x+3 . It holds � ( � ) ≥ 0 f(x)≥0 in:
To find where the function � ( � ) ≥ 0 f(x)≥0, we need to determine where the numerator and denominator have the same sign. � ( � )
� + 3 � − 1 f(x)= x−1 x+3
� + 3 ≥ 0 x+3≥0 implies � ≥ − 3 x≥−3 � − 1
0 x−1>0 implies �
1 x>1 � − 1 < 0 x−1<0 implies � < 1 x<1 Combining these:
� + 3 � − 1 ≥ 0 x−1 x+3 ≥0 in ( − ∞ , − 3 ] ∪ [ 1 , ∞ ) (−∞,−3]∪[1,∞) Therefore, the correct answer is ( − ∞ , − 3 ] ∪ [ 1 , ∞ ) (−∞,−3]∪[1,∞).
Solution
It looks like you're analyzing the rational function:
and determining the intervals where .
The reasoning you've laid out involves finding the intervals where the numerator and the denominator have the same sign (both positive or both negative), since a fraction is non-negative when both parts are either both positive or both negative.
Steps:
-
Numerator :
- The numerator is zero at .
- It is positive for , and negative for .
-
Denominator :
- The denominator is zero at .
- It is positive for , and negative for .
To find the solution, we need to examine the signs of the numerator and denominator in the intervals determined by the critical points and :
-
Interval 1:
- The numerator is negative, and the denominator is negative, so the fraction is positive.
-
Interval 2:
- The numerator is negative, and the denominator is positive, so the fraction is negative.
-
Interval 3:
- The numerator is positive, and the denominator is positive, so the fraction is positive.
At , the function is undefined due to division by zero, so we exclude from the solution. At , the function is , so is included in the solution.
Final Solution:
The solution is where , which is:
Questions for further exploration:
- What is the behavior of the function as approaches the asymptotes at ?
- How would the solution change if the inequality was instead of ?
- Can you find the vertical asymptote and horizontal asymptote of the function ?
- How would the graph of this function look for the intervals and ?
- What would happen to the function if you added or subtracted a constant to the numerator or denominator?
Tip:
When solving inequalities involving rational functions, always carefully consider where the function is undefined (denominator equals zero), and check intervals between critical points where the numerator and denominator change sign.
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Math Problem Analysis
Mathematical Concepts
Rational Functions
Inequalities
Sign Analysis
Formulas
f(x) = (x - 1) / (x + 3)
Theorems
Sign analysis of rational functions
Critical points and intervals of a rational function
Suitable Grade Level
Grades 9-11