Range of the Function f(x) (x^2 - 2x - 3) / (x^2 5x 4)

In this article, we will explore the range of the function f(x) (x^2 - 2x - 3) / (x^2 5x 4). Understanding the range of a function is crucial for many applications in mathematics, engineering, and other fields. We will break down the solution into steps, making it accessible and easy to follow.

Introduction to the Function

The given function is:

y (x^2 - 2x - 3) / (x^2 5x 4)

Step 1: Analyzing the Function

First, we need to understand the behavior of the function. The numerator is a quadratic equation, and the denominator can also be factored. Let's factor both the numerator and the denominator to simplify the function:

1. Numerator: x^2 - 2x - 3

This can be factored as: (x - 3)(x 1)

2. Denominator: x^2 5x 4

This can be factored as: (x 1)(x 4)

Step 2: Simplifying the Function

Now, we can write the function in a simpler form:

y (x - 3)(x 1) / (x 1)(x 4)

However, we need to consider the domain of the function. The function is undefined where the denominator is zero. Therefore:

y (x - 3) / (x 4), where x ≠ -1

This simplification removes the common factor (x 1), but we must still consider x ≠ -1 as a restriction on the domain.

Step 3: Determining the Range

To find the range, we need to consider the following:

1. Horizontal Asymptote

As x approaches infinity or negative infinity, the function y (x - 3) / (x 4) approaches y 1. Therefore, y 1 is a horizontal asymptote.

lim x -> ∞ (x - 3) / (x 4) 1

lim x -> -∞ (x - 3) / (x 4) 1

2. Vertical Asymptote and Discontinuity

The function has a vertical asymptote at x -4 because the denominator becomes zero at this point. There is also a vertical discontinuity at x -1.

3. Critical Points

To find critical points, we take the derivative of the function and set it to zero:

y' d[(x - 3) / (x 4)] / dx

y' (x 4) - (x - 3) * 1 / (x 4)^2

y' (x 4)^2 - (x - 3) * (x 4) / (x 4)^2

y' (x^2 8x 16 - x^2 x - 12) / (x 4)^2

y' (9x 4) / (x 4)^2

Solving for y' 0:

9x 4 0

x -4/9

4. Analysis of the Function

Now, we analyze the function around the critical points and asymptotes:

When x -4/9:

y (-4/9 - 3) / (-4/9 4)

y (-4/9 - 27/9) / (36/9 - 4/9)

y (-31/9) / (32/9)

y -31/32

When x approaches -4 from the left, y approaches negative infinity, and when x approaches -4 from the right, y approaches positive infinity.

5. Conclusion on Range

The range of the function y (x - 3) / (x 4) is all real numbers except for the horizontal asymptote value of 1. Therefore:

Range: y ∈ (-∞, 1) U (1, ∞)

Conclusion

In conclusion, the range of the function f(x) (x^2 - 2x - 3) / (x^2 5x 4) is all real numbers except for the value 1. This detailed analysis helps us understand the behavior of the function and its limitations.

Key Concepts

The key concepts covered include:

1. Quadratic Equations

The numerator and denominator of the original function are quadratic equations, and factoring them simplifies the analysis.

2. Horizontal Asymptotes

The horizontal asymptote provides insight into the behavior of the function as x approaches infinity or negative infinity.

3. Vertical Asymptotes and Discontinuities

The vertical asymptotes and discontinuities at specific points help determine the range of the function.