Rational Function Examples With Answers

Understanding Rational Functions: Examples and Solutions

Rational functions are a fundamental concept in algebra, forming the backbone for many advanced mathematical concepts. Understanding them is crucial for success in higher-level mathematics and related fields like engineering and physics. This comprehensive guide explores various examples of rational functions, providing detailed solutions and explanations to solidify your understanding. We'll cover everything from basic definitions to more complex scenarios, ensuring you gain a strong grasp of this important topic.

What is a Rational Function?

A rational function is simply a function that can be expressed as the quotient of two polynomial functions. In other words, it's a fraction where both the numerator and the denominator are polynomials. The general form is:

f(x) = P(x) / Q(x)

where P(x) and Q(x) are polynomials, and Q(x) ≠ 0 (since division by zero is undefined).

Basic Examples and Solutions:

Let's start with some simple examples to build our understanding.

Example 1:

f(x) = (x + 2) / (x - 3)

This is a basic rational function. The numerator is a linear polynomial (x + 2), and the denominator is also a linear polynomial (x - 3). To find the value of the function at a specific point, simply substitute the x-value into the function.

• Find f(4): f(4) = (4 + 2) / (4 - 3) = 6 / 1 = 6

• Find f(0): f(0) = (0 + 2) / (0 - 3) = 2 / -3 = -2/3

• Find f(3): f(3) is undefined because the denominator becomes zero (division by zero is undefined). This indicates a vertical asymptote at x = 3.

Example 2:

f(x) = (x² + 3x + 2) / (x + 1)

Here, the numerator is a quadratic polynomial, and the denominator is a linear polynomial. Notice that the numerator can be factored:

f(x) = (x + 1)(x + 2) / (x + 1)

If x ≠ -1, we can cancel the (x + 1) terms, simplifying the function to:

f(x) = x + 2 (for x ≠ -1)

This simplification reveals that the graph of this rational function is identical to the line y = x + 2, except for a hole at x = -1. This hole represents the point where the original function is undefined.

Example 3:

f(x) = 2x / (x² - 4)

In this example, we have a linear polynomial in the numerator and a quadratic polynomial in the denominator. The denominator can be factored:

f(x) = 2x / [(x - 2)(x + 2)]

This function has vertical asymptotes at x = 2 and x = -2, where the denominator equals zero.

Analyzing Rational Functions: Key Features

Understanding the key features of rational functions is crucial for sketching their graphs and solving related problems. These features include:

• Domain: The set of all possible x-values for which the function is defined. This excludes values that make the denominator zero.
• Vertical Asymptotes: Vertical lines (x = a) where the function approaches positive or negative infinity as x approaches 'a'. They occur when the denominator is zero and the numerator is not zero.
• Horizontal Asymptotes: Horizontal lines (y = b) that the function approaches as x approaches positive or negative infinity. The existence and value of a horizontal asymptote depend on the degrees of the numerator and denominator polynomials.
• Oblique (Slant) Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. They represent the slanted line the function approaches as x approaches infinity or negative infinity.
• x-intercepts (zeros): The x-values where the function crosses the x-axis (i.e., where f(x) = 0). These occur when the numerator is zero and the denominator is not zero.
• y-intercept: The y-value where the function crosses the y-axis (i.e., where x = 0). This is found by evaluating f(0).
• Holes: Points where the function is undefined due to a common factor in the numerator and denominator that cancels out.

More Complex Examples and Solutions:

Let's delve into more complex scenarios:

Example 4:

f(x) = (x³ - 8) / (x² - 4)

The numerator can be factored using the difference of cubes formula: x³ - 8 = (x - 2)(x² + 2x + 4). The denominator can be factored as (x - 2)(x + 2).

f(x) = (x - 2)(x² + 2x + 4) / [(x - 2)(x + 2)]

For x ≠ 2, we can cancel the (x - 2) terms:

f(x) = (x² + 2x + 4) / (x + 2) (for x ≠ 2)

This function has a vertical asymptote at x = -2 and a hole at x = 2. To find the y-coordinate of the hole, substitute x = 2 into the simplified function: (2² + 2(2) + 4) / (2 + 2) = 12 / 4 = 3. So the hole is at the point (2, 3). It also has an oblique asymptote. To find it, perform polynomial long division:

          x + 0
x + 2 | x² + 2x + 4
          x² + 2x
              0 + 4

The oblique asymptote is y = x.

Example 5:

f(x) = (x² + 1) / (x⁴ + 1)

This function has no vertical asymptotes because the denominator is always positive and never zero. It does have a horizontal asymptote at y = 0 because the degree of the denominator is greater than the degree of the numerator.

Example 6: A Real-World Application

Consider a scenario involving the efficiency of a machine. Let's say the efficiency (E) of a machine is modeled by the rational function:

E(x) = (100x) / (x² + 10x + 25)

where x represents the amount of raw material used.

We can find the efficiency at different levels of raw material:

• If x = 5, E(5) = (100 * 5) / (25 + 50 + 25) = 500 / 100 = 5. This means that with 5 units of raw material, the efficiency is 5 units. This is not realistic. The model likely needs adjustments.

• If x = 10, E(10) = (100 * 10) / (100 + 100 + 25) = 1000 / 225 ≈ 4.44.

This example demonstrates how rational functions can model real-world phenomena, requiring careful interpretation of the results within the context of the problem.

Frequently Asked Questions (FAQ)

• How do I find the vertical asymptotes of a rational function? Set the denominator equal to zero and solve for x. The solutions (excluding any that also make the numerator zero) represent the vertical asymptotes.

• How do I find the horizontal asymptote? Compare the degrees of the numerator and denominator polynomials.

  • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
  • If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator).
  • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote, but there might be an oblique asymptote.

• How do I find the oblique asymptote? Perform polynomial long division of the numerator by the denominator. The quotient (ignoring the remainder) represents the equation of the oblique asymptote.

• What is a hole in a rational function? A hole occurs when there's a common factor in both the numerator and denominator that can be canceled out. The x-value that makes this common factor zero represents the x-coordinate of the hole. Substitute this x-value into the simplified function to find the y-coordinate of the hole.

Conclusion:

Rational functions are powerful tools for modeling various phenomena and solving complex mathematical problems. By understanding their key features – domain, vertical and horizontal asymptotes, oblique asymptotes, x-intercepts, y-intercepts, and holes – you can effectively analyze and graph these functions. This guide provides a solid foundation for further exploration of this essential topic in algebra and beyond. Remember to practice solving different types of rational function problems to solidify your understanding and build confidence in tackling more advanced concepts. The more examples you work through, the more proficient you will become in identifying and solving these problems.