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Analyzing the Rational Function: (x-1)/(x^2-1)
This article explores the properties and characteristics of the rational function (x-1)/(x^2-1), providing insights into its behavior, domain, range, asymptotes, and graphical representation.
Simplifying the Expression
The function can be simplified by factoring the denominator:
(x-1)/(x^2-1) = (x-1)/((x+1)(x-1))
Notice that there's a common factor of (x-1) in both the numerator and denominator. Therefore, we can cancel this factor, resulting in:
(x-1)/(x^2-1) = 1/(x+1), x ≠ 1
It's crucial to remember the restriction x ≠ 1 because we cancelled out the (x-1) term. This means the original function is undefined at x = 1.
Domain and Range
Domain:
The function is defined for all real numbers except for x = -1 and x = 1. This is because these values make the denominator zero, leading to an undefined expression. Therefore, the domain of the function is:
(-∞, -1) U (-1, 1) U (1, ∞)
Range:
The simplified function, 1/(x+1), can take on any real value except for 0. This is because the numerator is always 1, and the denominator can never be zero. Therefore, the range of the function is:
(-∞, 0) U (0, ∞)
Asymptotes
Vertical Asymptote:
The vertical asymptote occurs at x = -1, which is where the denominator of the simplified function becomes zero. This indicates that the function approaches infinity as x approaches -1.
Horizontal Asymptote:
The horizontal asymptote can be determined by analyzing the degrees of the numerator and denominator. In the simplified function, the degree of the denominator (1) is greater than the degree of the numerator (0). This means the horizontal asymptote is y = 0.
Graphing the Function
To visualize the function, consider the following:
- Holes: The function has a hole at x = 1 due to the cancelled factor.
- Vertical Asymptote: The vertical asymptote at x = -1 divides the graph into two sections.
- Horizontal Asymptote: The horizontal asymptote at y = 0 provides a boundary for the graph.
- Behavior near Asymptotes: As x approaches -1, the function approaches positive or negative infinity depending on the side of the asymptote. As x approaches positive or negative infinity, the function approaches 0.
The graph of the function will show a curve approaching the vertical and horizontal asymptotes, with a hole at x = 1.
Conclusion
Understanding the properties of the rational function (x-1)/(x^2-1) provides valuable insights into its behavior and graphical representation. By simplifying the expression, analyzing the domain and range, identifying asymptotes, and considering the behavior near these asymptotes, we can accurately visualize and comprehend the function's characteristics.