The function f(x) x/x is a fascinating case in understanding domain and range. This article delves into the analysis of the domain, range, and the significance of this function in relation to other mathematical concepts such as the signum function.
Domain of the Function
When examining the function f(x) x/x, we first consider its domain. The function is undefined when the denominator is zero, thus x cannot be zero. Therefore, the domain of f(x) x/x is all real numbers except zero ((R - {0})).
Behavior of the Function
For Positive and Negative Values of x
Let's consider the behavior of the function for specific values of x:
If x 0, then x x, hence f(x) x/x 1. If x 0, then x -x, hence f(x) -x/x -1. If x 0, then f(x) 0/0, which is undefined.Range of the Function
The range of the function f(x) x/x is determined by the output values of the function. Based on the behavior described above:
For x 0, f(x) 1. For x 0, f(x) -1. For x 0, f(x) is undefined.Therefore, the range of f(x) x/x is the set {-1, 1}. This can also be denoted as:
Range: ({-1, 1})
Signum Function and Reciprocal Function
The signum function, denoted as sgn(x), is defined as:
sgn(x) 1 if x 0. sgn(x) -1 if x 0. sgn(x) 0 if x 0.The reciprocal of the signum function, denoted as 1/|x|, is a similar function but only defined for x ≠ 0:
1/|x| 1 if x 0. 1/|x| -1 if x 0. 1/|x| is undefined if x 0.Therefore, the domain of the reciprocal of the signum function is also R - {0} and the range is {-1, 1}.
Practical Implications and Continuity
The function f(x) x/x is not continuous at x 0 due to its undefined value. However, it is continuous for all other values of x in the domain. This discontinuity at x 0 affects its range.
In practical applications, understanding the domain and range of such functions is crucial for correctly interpreting and utilizing the data. For instance, in graph theory, operations, and algorithms involving limits and derivatives, understanding these concepts ensures accurate results.
Conclusion
In conclusion, the function f(x) x/x has a domain of all real numbers except zero and a range of {-1, 1}. Understanding the domain and range is essential in many areas of mathematics, including calculus and the analysis of continuity. The concept is closely related to the signum function and its reciprocal, providing a deeper insight into such mathematical functions.
By grasping these fundamental concepts, one can effectively analyze and utilize functions in various mathematical and practical contexts.