Understanding the Domain and Range of the Function y (x-1)/x
In the study of mathematical functions, it is crucial to identify the domain and range to fully understand the behavior and limitations of the function. This article delves into the domain and range analysis of the function y (x-1)/x, providing a comprehensive guide for students and professionals alike.
Introduction to the Function y (x-1)/x
The given function, y (x-1)/x, is a rational function where the numerator is linear and the denominator is a monomial. Rational functions such as these are essential in algebra, calculus, and various scientific applications. Understanding their domain and range is fundamental for interpreting graphical representations and solving real-world problems.
Domain of the Function
The domain of a function refers to all possible input values for which the function is defined. For the function y (x-1)/x, the domain is determined by identifying any values of x for which the function is undefined:
The function y (x-1)/x will be undefined when the denominator is zero. Therefore, the domain is all real numbers x such that x ≠ 0.Making this clear, the domain of the function is (mathbb{R} - {0}), where (mathbb{R}) represents the set of all real numbers.
To visualize the domain, we can plot the function on a graph. The function will have a vertical asymptote at (x 0) because the denominator (x) becomes zero. This absence of a value at (x 0) is what limits the domain to all real numbers except zero.
Range of the Function
The range of a function is the set of all possible output values. To find the range of y (x-1)/x, we analyze the behavior of the function as x varies:
When (x 1), the function is undefined because the numerator and denominator both approach zero, which may lead to an indeterminate form. However, factoring and simplifying can help us understand the function's behavior around this the limit as (x) approaches infinity or negative infinity. As (x) becomes very large or very small, the term (1/x) approaches zero. Thus, the function behaves like (y 1). However, the function will never reach (y 1) exactly unless (x 1).To explore the exact range, we can rewrite the function in a more revealing form:y (x-1)/x 1 - 1/x
This form reveals that the term (-1/x) can take on any value from (-infty) to (infty), except when it equals zero. Thus, (y) can take on all real values except (y 1).
Therefore, the range of the function is all real numbers except (y 1). Mathematically, the range can be expressed as (mathbb{R} - {1}).
Graphically, this can be visualized as a horizontal asymptote at (y 1), indicating that the function approaches this value but never reaches it.
Graphical Interpretation of the Function
The graph of the function y (x-1)/x will help us understand the domain and range more clearly:
In the graph, we can observe the vertical asymptote at (x 0) and the horizontal asymptote at (y 1). These asymptotes define the limitations within the domain and range, respectively.
Conclusion
The function y (x-1)/x has a domain of all real numbers except (x 0) and a range of all real numbers except (y 1). Understanding these properties is crucial for analyzing the function's behavior and interpreting its graphical representation.