Identifying Rational Functions From Zeros A Comprehensive Guide

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In mathematics, a rational function is defined as a function that can be expressed as the quotient of two polynomials. Understanding the behavior of these functions, especially their zeros, is crucial in various mathematical contexts. The zeros of a function are the values of x for which the function equals zero. In this article, we will explore how to identify the expression of a rational function given its zeros. Specifically, we will address the question: The zeros of a rational function g are -4 and 3. Which of the following expressions could define g(x)?

To answer this question effectively, we will delve into the fundamental principles of rational functions, focusing on the relationship between zeros and factors, and how these factors appear in the numerator and denominator of a rational function. By the end of this guide, you will have a solid understanding of how to determine the correct expression for a rational function based on its zeros, and you will be equipped to tackle similar problems with confidence.

Defining Rational Functions and Their Zeros

To begin, let's clearly define what a rational function is. A rational function is any function that can be written in the form g(x) = P(x) / Q(x), where P(x) and Q(x) are polynomial functions, and Q(x) is not equal to zero. Polynomial functions themselves are expressions consisting of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. Examples of polynomial functions include x² + 3x - 2 and 5x⁴ - 1.

The zeros of a rational function are the values of x for which the function g(x) equals zero. In other words, these are the solutions to the equation P(x) / Q(x) = 0. For a fraction to be zero, the numerator must be zero while the denominator must not be zero. Therefore, the zeros of a rational function are the zeros of the numerator P(x), provided that these values do not also make the denominator Q(x) zero. If a value makes both the numerator and the denominator zero, it may represent a hole (a removable discontinuity) in the graph of the function, rather than a simple zero.

Understanding this fundamental concept is key to identifying the expression of a rational function based on its zeros. The zeros directly correspond to factors in the numerator of the rational function, which we will explore in more detail in the next section.

The Relationship Between Zeros and Factors

The zeros of a function are intimately connected to its factors. If a number r is a zero of a function P(x), then (x - r) is a factor of P(x). This principle is a cornerstone of polynomial algebra and is crucial in understanding rational functions.

Consider a polynomial function P(x). If r₁, r₂, ..., rn are the zeros of P(x), then P(x) can be written in the factored form:

P(x) = a(x - r₁)(x - r₂)...(x - rn)

where a is a constant. This form explicitly shows the relationship between the zeros and the factors of the polynomial. Each factor (x - ri) corresponds to a zero ri. When x = ri, the factor (x - ri) becomes zero, making the entire polynomial P(x) equal to zero.

Now, let's apply this concept to rational functions. If g(x) = P(x) / Q(x) is a rational function, the zeros of g(x) are the zeros of P(x) that are not also zeros of Q(x). Therefore, if r is a zero of g(x), then (x - r) must be a factor of P(x). This is a crucial piece of information when trying to determine the expression of a rational function given its zeros. We can construct the numerator of the rational function by including factors corresponding to the given zeros.

In our problem, the zeros of the rational function g are -4 and 3. This means that (x - (-4)) and (x - 3), which simplify to (x + 4) and (x - 3), respectively, must be factors of the numerator P(x). This narrows down the possible expressions for g(x) significantly, allowing us to focus on options that include these factors in the numerator.

Analyzing the Given Options

Now, let's consider the given options for the expression of g(x), keeping in mind that the numerator must contain the factors (x + 4) and (x - 3):

A.

(x-2)(x+5) / (x-3)(x+4)

B.

(x-3)(x+4) / (x-2)(x+5)

By examining these options, we can immediately see which one aligns with the requirement that the numerator contains the factors (x + 4) and (x - 3). Option B has the factors (x - 3) and (x + 4) in the numerator, which correspond to the zeros 3 and -4, respectively. Option A, on the other hand, has these factors in the denominator, which would correspond to vertical asymptotes rather than zeros.

Therefore, based on our understanding of the relationship between zeros and factors, and after analyzing the given options, we can confidently identify the correct expression for g(x).

Determining the Correct Expression

Based on our analysis, option B is the only expression that contains the factors (x - 3) and (x + 4) in the numerator. This confirms that the zeros of the function are indeed 3 and -4. The denominator (x - 2)(x + 5) does not affect the zeros of the function; instead, it determines the vertical asymptotes, which occur at x = 2 and x = -5.

Therefore, the correct expression for g(x) is:

g(x) = (x-3)(x+4) / (x-2)(x+5)

This expression satisfies the condition that the zeros of g(x) are -4 and 3. To further verify, we can set the numerator equal to zero and solve for x:

(x - 3)(x + 4) = 0

This equation has two solutions: x = 3 and x = -4, which are the zeros we were given.

By carefully analyzing the relationship between zeros and factors, and by examining the given options, we were able to successfully determine the correct expression for the rational function g(x).

Key Takeaways and Further Exploration

In this article, we have explored the crucial concept of zeros in rational functions and how they relate to the factors of the numerator. We learned that if r is a zero of a rational function, then (x - r) is a factor of the numerator. This understanding allowed us to identify the correct expression for a rational function given its zeros.

Here are some key takeaways from our discussion:

  1. Rational functions are expressed as the quotient of two polynomials, g(x) = P(x) / Q(x).
  2. The zeros of a rational function are the values of x for which the function equals zero, which are the zeros of the numerator P(x) that are not also zeros of the denominator Q(x).
  3. If r is a zero of a function, then (x - r) is a factor of the polynomial.
  4. To determine the expression of a rational function given its zeros, identify the factors corresponding to the zeros and ensure they are in the numerator.

To further enhance your understanding of rational functions, consider exploring the following topics:

  • Vertical asymptotes: These occur at the zeros of the denominator, where the function becomes undefined.
  • Holes (removable discontinuities): These occur when a factor is present in both the numerator and the denominator.
  • Horizontal asymptotes: These describe the behavior of the function as x approaches positive or negative infinity.
  • Graphing rational functions: Understanding the zeros, asymptotes, and holes is crucial for accurately graphing rational functions.

By delving deeper into these topics, you will gain a more comprehensive understanding of rational functions and their properties. This knowledge will be invaluable in various mathematical and scientific contexts.

In conclusion, understanding the zeros of rational functions is a fundamental skill in mathematics. By grasping the relationship between zeros and factors, and by carefully analyzing the components of a rational function, you can confidently identify and work with these functions. Remember to always consider the context of the problem and apply the principles we have discussed to arrive at the correct solution. With practice and a solid understanding of these concepts, you will be well-equipped to tackle a wide range of problems involving rational functions.