Math Problem Statement

answer

The function r is given by r(x) = (x^2 - x - 2) / ((x + 1)^2(x - 2)). In the xy-plane, which of the following is true about holes in the graph of r?

Solution

The function given in the image is:

r(x)=x2x2(x+1)2(x2)r(x) = \frac{x^2 - x - 2}{(x + 1)^2(x - 2)}

We are tasked with finding which statement about the holes in the graph of r(x)r(x) is correct.

Step-by-Step Solution:

  1. Factor the numerator: x2x2=(x2)(x+1)x^2 - x - 2 = (x - 2)(x + 1) So, the function can be rewritten as: r(x)=(x2)(x+1)(x+1)2(x2)r(x) = \frac{(x - 2)(x + 1)}{(x + 1)^2(x - 2)}

  2. Simplify the expression:

    • The factor (x+1)(x + 1) in the numerator can cancel with one of the (x+1)(x + 1) terms in the denominator, so we have: r(x)=1x+1,with restrictions.r(x) = \frac{1}{x + 1}, \quad \text{with restrictions}.
    • However, there are restrictions on the domain because of the original factors in the denominator. Specifically:
      • At x=1x = -1, the original denominator becomes zero due to the (x+1)2(x + 1)^2 term, which would cause a hole in the graph.
      • At x=2x = 2, both the numerator and denominator have a factor of (x2)(x - 2), which indicates another hole at x=2x = 2.
  3. Multiplicity Analysis:

    • For x=1x = -1, the factor (x+1)(x + 1) appears twice in the denominator (with a power of 2) and once in the numerator. This means the multiplicity of x=1x = -1 is greater in the denominator, leading to a hole.
    • For x=2x = 2, the factor (x2)(x - 2) appears both in the numerator and denominator with equal multiplicities, indicating a hole as well.

Conclusion:

There are holes at both x=1x = -1 and x=2x = 2 because:

  • The multiplicity of x=1x = -1 in the denominator is greater than in the numerator.
  • The multiplicity of x=2x = 2 in the numerator and denominator is equal.

Thus, the correct answer is Option A.


Would you like a detailed explanation of any of these steps? Here are some related questions for deeper understanding:

  1. How do you find holes in the graph of a rational function?
  2. What is the significance of multiplicity in determining the nature of a hole?
  3. How do you simplify rational expressions involving polynomials?
  4. What are the conditions for a vertical asymptote versus a hole in a graph?
  5. How do repeated factors in the denominator affect the graph of a rational function?

Tip: Always check for holes by simplifying the function and examining common factors in the numerator and denominator before and after simplification.

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Graphing
Multiplicity
Holes in Graphs

Formulas

Factoring Polynomials
Simplifying Rational Expressions

Theorems

Multiplicity and Holes in Rational Functions

Suitable Grade Level

Grades 10-12