Question

The domain of a one-to-one function \(f\) is \([5, \infty),\) and its range is \([-2, \infty)\). State the domain and the range of \(f^{-1}\).

Short answer

The domain of \(f^{-1}\) is \([-2, \, \infty)\), and the range of \(f^{-1}\) is \([5, \, \infty)\).

Step-by-step solution

Understanding the Inverse Function

The inverse function, denoted as \(f^{-1}\), effectively reverses the roles of the domain and range of the original function \(f\). Therefore, the domain of \(f^{-1}\) is the range of \(f\), and the range of \(f^{-1}\) is the domain of \(f\).

Determine the Domain of \(f^{-1}\)

Since the range of \(f\) is \([-2, \, \infty)\), this becomes the domain of \(f^{-1}\).

Determine the Range of \(f^{-1}\)

Since the domain of \(f\) is \([5, \, \infty)\), this becomes the range of \(f^{-1}\).

State the Domain and Range of \(f^{-1}\)

Thus, the domain of \(f^{-1}\) is \([-2, \, \infty)\), and the range of \(f^{-1}\) is \([5, \, \infty)\).

Key concepts

Domain and Range

The domain and range are fundamental concepts when working with functions.
The domain of a function is the set of all possible input values. These inputs are typically the x-values on a graph.
The range is the set of all possible output values, or the y-values on a graph.
When dealing with inverse functions, the domain and range swap roles. The domain of the original function becomes the range of the inverse function, and vice versa.
In our example, the function \(f\) has a domain of \( [5, \, \infty) \) and a range of \( [-2, \, \infty) \). For the inverse function \(f^{-1}\), the domain will be \( [-2, \, \infty) \) and the range will be \( [5, \, \infty) \). This is because the inverse function essentially swaps the x and y values of the original function.

One-to-One Functions

A one-to-one function is a function where each output is associated with exactly one input. This means no two different inputs share the same output.
One-to-one functions are important because only they have inverse functions.
To determine if a function is one-to-one, you can use the Horizontal Line Test. If any horizontal line intersects the graph of a function more than once, then the function is not one-to-one.
In the given exercise, the function \(f\) is specified as one-to-one. This tells us that there is an inverse function \(f^{-1}\) without any complications. Thus, finding the domain and range for \(f^{-1}\) becomes straightforward.

Function Notation

Function notation provides a way to represent functions and their inverses concisely.
The notation \(f(x)\) signifies the output of the function \(f\) for an input \(x\).
The notation \(f^{-1}(x)\) signifies the output of the inverse function for an input \(x\).
Function notation is useful because it helps in understanding and distinguishing between the original function and its inverse.
For instance, in our problem, expressing the domain and range in function notation clarifies that the domain of \(f^{-1}\) is \( [-2, \, \infty) \) and its range is \( [5, \, \infty). \)