(x-2)(x-3)%2F(x-4)(x-5)(x-6).jpeg "(x-1)(x-2)(x-3)/(x-4)(x-5)(x-6)")
Exploring the Rational Function (x-1)(x-2)(x-3)/(x-4)(x-5)(x-6)
This article delves into the fascinating world of the rational function:
(x-1)(x-2)(x-3) / (x-4)(x-5)(x-6)
We'll explore its key features, including domain, intercepts, asymptotes, and behavior.
Understanding the Basics
First, let's break down the function. It's a rational function, meaning it's a ratio of two polynomials.
- The numerator is (x-1)(x-2)(x-3), which is a cubic polynomial.
- The denominator is (x-4)(x-5)(x-6), also a cubic polynomial.
Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined. A rational function is undefined when the denominator equals zero.
Therefore, the domain of this function is all real numbers except for x = 4, x = 5, and x = 6, as these values would make the denominator zero.
Intercepts
x-intercepts occur where the function crosses the x-axis, meaning the y-value is zero. This happens when the numerator is zero.
- The numerator is zero when x = 1, x = 2, or x = 3.
- Therefore, the x-intercepts are (1, 0), (2, 0), and (3, 0).
The y-intercept occurs where the function crosses the y-axis, meaning the x-value is zero.
- Substituting x = 0 into the function, we get (1)(2)(3) / (4)(5)(6) = 1/20.
- The y-intercept is (0, 1/20).
Asymptotes
Asymptotes are lines that the function approaches as x approaches positive or negative infinity.
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Vertical asymptotes occur at the values of x that make the denominator zero.
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This function has three vertical asymptotes: x = 4, x = 5, and x = 6.
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Horizontal asymptotes are determined by comparing the degrees of the numerator and denominator.
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Both the numerator and denominator have the same degree (3).
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The horizontal asymptote is the line y = the ratio of the leading coefficients of the numerator and denominator, which is y = 1.
Behavior
End behavior refers to how the function behaves as x approaches positive or negative infinity.
- Since the degree of the numerator and denominator is the same, the function approaches the horizontal asymptote (y = 1) as x approaches positive or negative infinity.
Local behavior refers to how the function behaves near its vertical asymptotes.
- The function will either approach positive or negative infinity as it gets closer to a vertical asymptote.
- To determine the exact behavior near each vertical asymptote, you can analyze the signs of the numerator and denominator in small intervals around each vertical asymptote.
Conclusion
This exploration has provided a comprehensive understanding of the function (x-1)(x-2)(x-3)/(x-4)(x-5)(x-6). By analyzing its domain, intercepts, asymptotes, and behavior, we can visualize its graph and predict its behavior over different intervals. This understanding can be applied to various applications, such as solving equations, modeling real-world phenomena, and exploring the relationship between input and output values.