Understanding Inverse Domain and Range for a Given Relation

In the context of relations and functions, the domain and range can be understood as the set of all input values (x) and the set of all output values (y) that a function can produce. When dealing with a specific relation such as R {xy: y x^2 1}, we need to explore the concept of domain, range, and inverses to fully understand its behavior.

Defining the Relation and Its Graph

The given relation R can be expressed as y x^2 1. This equation represents a parabola that opens upwards with its vertex at (0, 1).

The graph of this relation is a “bowl” shape, where every value of x produces a corresponding value for y, following the equation y x^2 1. This is a standard parabolic curve with its axis of symmetry along the y-axis (x 0).

Domain and Range of the Relation

The domain of a relation is the set of all possible input values (x). For the given relation y x^2 1, the domain is all real numbers (since there are no restrictions on x). We can express this as:

Domain: All real numbers (?∞, ∞)

The range of the relation is the set of all possible output values (y). For the equation y x^2 1, as x varies over all real numbers, y will always be greater than or equal to 1. Therefore, the range is all y such that y ≥ 1. We can express this as:

Range: {y : y ≥ 1}

Inverse of the Relation

When considering the inverse of a relation, we are looking for a function that reverses the original function. For y x^2 1, it is important to note that this function is not one-to-one. A function is one-to-one if each y-value corresponds to exactly one x-value, and vice versa.

In the case of y x^2 1, for each y-value greater than or equal to 1, there are two corresponding x-values: one positive and one negative (e.g., if y 2, x can be either √1 or -√1). This means that an inverse function does not exist because we cannot uniquely determine an x-value from a given y-value. In other words, the inverse would not be a function as it would fail the vertical line test.

Mathematically, this can be expressed as:

Inverse: Does not exist

Specifying Domains and Ranges

In the context of specifying domains and ranges, it's crucial to understand that you can choose the domain and range based on the nature of the problem or the context. For example, if the domain is restricted to real numbers, the domain is all real numbers, and the range is y ≥ 1. If the domain is restricted to a subset of real numbers, such as integers or natural numbers, then the domain and range can be adjusted accordingly.

For instance, if we consider x to be integers, then the domain would be all integers, and the range would be y ≥ 1 where y is also an integer. If x is limited to natural numbers, the domain would be all natural numbers, and the range would still be y ≥ 1.

It's important to note that the relation y x^2 1 is not defined for complex numbers because it involves the square root operation, which is not defined for complex numbers in a real-number context.

Conclusion

Understanding the domain, range, and inverse of a relation (such as y x^2 1) provides critical insights into the behavior and properties of the function. As we have seen, the relation y x^2 1 is a parabola that opens upwards with a vertex at (0, 1). Its domain and range are all real numbers and y ≥ 1, respectively. The inverse of this relation does not exist due to its non-one-to-one nature.