Introduction

Rational functions play a crucial role in precalculus and beyond. They are expressions formed by dividing one polynomial by another. Understanding these functions is important, as they appear in various scientific and engineering contexts. One key aspect of studying rational functions is finding their zeros. This means determining the values for which the function equals zero. Let’s explore this concept in detail and learn how to tackle 1.8 Rational Functions and Zeros effectively!

Understanding Rational Functions

A rational function is a ratio of two polynomials. For example, the function r(x) = \frac{x^2 - 4}{x - 2} has a numerator of x^2 - 4 and a denominator of x - 2. When dealing with rational functions, the domain is especially important. The domain consists of all x-values that don’t make the denominator zero. Therefore, the example function’s domain excludes x = 2. Recognizing these restrictions helps in accurately finding zeros.

What Are Zeros of a Rational Function?

Zeros of a rational function are the values of x that make the function equal to zero. To find the zeros, the numerator must be set to zero, provided the resulting x-value doesn’t violate the domain restrictions. The relationship between zeros and the numerator is straightforward: if the numerator is zero, the entire function equals zero, as long as the denominator isn’t zero as well.

Step-by-Step Guide on How to Find Zeros of a Rational Function

Step 1: Identify the Rational Function

Let’s work with r(x) = \frac{x^2 - 4}{x - 2} as an example.

Step 2: Set the Numerator Equal to Zero

The zeros of a rational function come from its numerator. Therefore:

• Given r(x) = \frac{x^2 - 4}{x - 2}
• Set the numerator, x^2 - 4, equal to zero: x^2 - 4 = 0

Step 3: Solve for X

Now, solve the equation:

• Factor the numerator: (x - 2)(x + 2) = 0
• Find the solutions: x = 2 and x = -2

Step 4: Check the Domain

It’s essential to ensure that the zeros are valid within the function’s domain.

• For the domain of r(x), x = 2 isn’t included since it makes the denominator zero (x - 2).

Step 5: Conclude with Valid Zeros

The valid zero for this function is x = -2. Therefore, while x = 2 is a zero of the numerator, it isn’t a zero of the function due to the domain restriction.

Rational Function Inequalities and How to Find Zeros of a Rational Function

Rational inequalities involve finding where a rational function is greater than or less than zero. These zeros often act as critical endpoints or asymptotes:

• Consider the inequality \frac{x^2 - 4}{x - 2} \geq 0.
• Using previously found zeros, analyze intervals around these points.
• Solving inequalities often requires testing intervals to determine where the inequality holds true.

Practice Problems

Here are a few practice problems to enhance understanding:

1. Find the zeros of f(x) = \frac{x^2 + x - 6}{x - 3}.
2. Solve the rational inequality \frac{x^2 - 1}{x - 3} > 0 .

Solutions:

For Problem 1:

• Set the numerator x^2 + x - 6 = 0.
• Factor to find (x - 2)(x + 3) = 0.
• The solutions are x = 2 and x = -3.
• Check these against the domain to see if they are valid.
  • Neither of these values makes the denominator equal to zero, so both values are zeros of the function.

For Problem 2:

• Factor the numerator: (x - 1)(x + 1) > 0.
• Test intervals created by critical points x = 1, -1, 3.
  • For (-\infty, -1), use x = -2:
    • \frac{(-2 - 1)(-2 + 1)}{-2 - 3} = \frac{(-3)(-1)}{-5} = \frac{3}{-5} < 0
  • For (-1, 1), use x = 0:
    • \frac{(0 - 1)(0 + 1)}{0 - 3} = \frac{(-1)(1)}{-3} = \frac{-1}{-3} > 0
  • For (1, 3), use x = 2:
    • \frac{(2 - 1)(2 + 1)}{2 - 3} = \frac{(1)(3)}{-1} = \frac{3}{-1} < 0
  • For (3, \infty), use x = 4:
    • \frac{(4 - 1)(4 + 1)}{4 - 3} = \frac{(3)(5)}{1} = 15 > 0
• Determine where the product of intervals is positive.
  • The inequality is satisfied when \frac{(x - 1)(x + 1)}{x - 3} > 0. From the test results, this occurs in the intervals:
    • (-1, 1) \cup (3, \infty).

Quick Reference Chart: Vocabulary and Definitions

Conclusion

How to find zeros of a rational function is crucial in precalculus. This knowledge assists in graphing functions and solving inequalities. Regular practice is recommended for mastery. Further resources can aid in deepening this understanding. Engage in discussions or study groups to share strategies for solving these functions and problems.

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