Find The Values Of $x$ For Which The Denominator Is Equal To Zero For $y=\frac{x-3}{x^2-1}$.A. $ X = − 1 X = -1 X = − 1 [/tex]B. $x = -1, X = 1$C. NoneD. $x = 1, X = 3$

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Solving for the Values of x in a Rational Function

In mathematics, rational functions are a type of function that can be expressed as the ratio of two polynomials. These functions are commonly used in various mathematical and scientific applications. However, when working with rational functions, it's essential to be aware of the values of the variable that make the denominator equal to zero, as these values are not part of the function's domain. In this article, we will explore how to find the values of x for which the denominator is equal to zero in the rational function y = (x - 3) / (x^2 - 1).

Understanding the Rational Function

The given rational function is y = (x - 3) / (x^2 - 1). To find the values of x that make the denominator equal to zero, we need to set the denominator, x^2 - 1, equal to zero and solve for x.

Setting the Denominator Equal to Zero

To find the values of x that make the denominator equal to zero, we set x^2 - 1 = 0 and solve for x.

x^2 - 1 = 0

Solving the Quadratic Equation

The equation x^2 - 1 = 0 is a quadratic equation, which can be solved using various methods, including factoring, the quadratic formula, or completing the square.

Factoring the Quadratic Equation

We can factor the quadratic equation x^2 - 1 = 0 as follows:

(x + 1)(x - 1) = 0

Solving for x

To find the values of x that satisfy the equation (x + 1)(x - 1) = 0, we set each factor equal to zero and solve for x.

x + 1 = 0 --> x = -1
x - 1 = 0 --> x = 1

In conclusion, the values of x that make the denominator equal to zero in the rational function y = (x - 3) / (x^2 - 1) are x = -1 and x = 1. These values are not part of the function's domain, and the function is undefined at these points.

The correct answer is B. x = -1, x = 1.

  • When working with rational functions, it's essential to be aware of the values of the variable that make the denominator equal to zero.
  • To find the values of x that make the denominator equal to zero, set the denominator equal to zero and solve for x.
  • The values of x that make the denominator equal to zero are not part of the function's domain.
  • Rational functions can be used to model various real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of electrical circuits.

Rational functions have numerous real-world applications, including:

  • Physics and Engineering: Rational functions are used to model the motion of objects, the growth of populations, and the behavior of electrical circuits.
  • Economics: Rational functions are used to model the behavior of economic systems, including the supply and demand of goods and services.
  • Biology: Rational functions are used to model the growth of populations, the spread of diseases, and the behavior of ecosystems.

Frequently Asked Questions

Q: What is a rational function?

A: A rational function is a type of function that can be expressed as the ratio of two polynomials. It is a function that can be written in the form y = f(x) / g(x), where f(x) and g(x) are polynomials.

Q: What is the domain of a rational function?

A: The domain of a rational function is the set of all values of x for which the denominator is not equal to zero. In other words, the domain of a rational function is the set of all values of x that make the denominator nonzero.

Q: How do I find the values of x that make the denominator equal to zero?

A: To find the values of x that make the denominator equal to zero, set the denominator equal to zero and solve for x. This can be done using various methods, including factoring, the quadratic formula, or completing the square.

Q: What happens when the denominator is equal to zero?

A: When the denominator is equal to zero, the function is undefined at that point. In other words, the function is not defined for the values of x that make the denominator equal to zero.

Q: Can I simplify a rational function?

A: Yes, you can simplify a rational function by canceling out any common factors between the numerator and the denominator. This can be done using various methods, including factoring, the quadratic formula, or completing the square.

Q: How do I graph a rational function?

A: To graph a rational function, you can use various methods, including plotting points, using a graphing calculator, or using a computer algebra system. You can also use the following steps:

  1. Find the values of x that make the denominator equal to zero.
  2. Plot these values on the graph.
  3. Determine the behavior of the function as x approaches positive and negative infinity.
  4. Use this information to sketch the graph of the function.

Q: What are some common types of rational functions?

A: Some common types of rational functions include:

  • Linear rational functions: These are rational functions of the form y = ax + b / cx + d, where a, b, c, and d are constants.
  • Quadratic rational functions: These are rational functions of the form y = ax^2 + bx + c / dx^2 + ex + f, where a, b, c, d, e, and f are constants.
  • Polynomial rational functions: These are rational functions of the form y = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0 / b_m x^m + b_(m-1) x^(m-1) + ... + b_1 x + b_0, where a_i and b_i are constants.

Q: How do I use rational functions in real-world applications?

A: Rational functions can be used to model various real-world phenomena, including:

  • Physics and Engineering: Rational functions can be used to model the motion of objects, the growth of populations, and the behavior of electrical circuits.
  • Economics: Rational functions can be used to model the behavior of economic systems, including the supply and demand of goods and services.
  • Biology: Rational functions can be used to model the growth of populations, the spread of diseases, and the behavior of ecosystems.

In conclusion, rational functions are a powerful tool for modeling various real-world phenomena. By understanding the properties and behavior of rational functions, you can use them to solve a wide range of problems in physics, engineering, economics, and biology.