Explain how the domain and range of a one to one function relate to the domain and range of its inverse function. Use grammatically correct complete sentences
The domain and range of a one-to-one function and its inverse function are related in a unique way. In a one-to-one function, each input value in the domain maps to a unique output value in the range, and no two different input values can map to the same output value. Consequently, in the inverse function, the roles of the domain and range are switched. The domain of the inverse function consists of the range values of the original function, while the range of the inverse function encompasses the domain values of the original function. This means that the domain and range of a one-to-one function and its inverse function are interchanged.
The domain of a one-to-one function is the set of input values for which each unique element in the domain corresponds to a unique element in the range. In other words, every input has a unique output. The range of a one-to-one function, on the other hand, is the set of output values that correspond to the unique input values.
When we find the inverse of a one-to-one function, the roles of the domain and range are reversed. The domain of the inverse function will be the range of the original function, and the range of the inverse function will be the domain of the original function. This is because the inverse function essentially switches the inputs and outputs of the original function.
Therefore, if a one-to-one function has a certain domain, its inverse function will have the same set of values in its range. Similarly, if the original function has a particular range, the inverse function will have the same set of values in its domain.
In conclusion, the domain and range of a one-to-one function and its inverse function are interchanged, with the domain becoming the range and the range becoming the domain.
The domain and range of a one-to-one function and its inverse function are interconnected in a special way. Let's start with the domain. In a one-to-one function, every input (or x-value) is paired with a unique output (or y-value), meaning no two different inputs can produce the same output. Consequently, the domain of the original function is equal to the range of its inverse function.
Now, let's consider the range. In a one-to-one function, every output value is also unique since no two different inputs can generate the same output. Therefore, the range of the original function corresponds to the domain of its inverse function.
In summary, the domain of a one-to-one function is the same as the range of its inverse function, and the range of the one-to-one function is equivalent to the domain of its inverse function.