Question

The domain of a one-to-one function \(f\) is \([0, \infty),\) and its range is \([5, \infty)\). State the domain and the range of \(f^{-1}\).

Short answer

Domain: \([5, \infty)\). Range: \([0, \infty)\).

Step-by-step solution

Understand the Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. Here, for the function \(f\), the domain is given as \([0, \infty)\), which means the function can take any real number starting from 0 and increasing to infinity. The range of \(f\) is given as \([5, \infty)\), meaning the outputs of the function start at 5 and go up to infinity.

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

The domain of the inverse function \(f^{-1}\) is the range of the original function \(f\). Since the range of \(f\) is \([5, \infty)\), the domain of \(f^{-1}\) will be \([5, \infty)\).

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

The range of the inverse function \(f^{-1}\) is the domain of the original function \(f\). Since the domain of \(f\) is \([0, \infty)\), the range of \(f^{-1}\) will be \([0, \infty)\).

Conclude the Solution

Based on our findings, the domain of \(f^{-1}\) is \([5, \infty)\) and the range of \(f^{-1}\) is \([0, \infty)\).

Key concepts

Domain and Range

The domain and range are fundamental concepts in understanding functions. They tell us the 'inputs' and 'outputs' a function can handle. For a function, the domain is the set of values you can put in, and the range is what you get out. Let's look at an example to make it clear. If you have a function \( f \) with a domain of \([0, \infty)\) and a range of \([5, \infty)\), it means:

• The function can only take input values starting from 0 to infinity.
• It will produce output values starting from 5 to infinity.

So, understanding the domain and range lets you know exactly what values a function works with, which is crucial for solving many types of problems.

Inverse Function

An inverse function essentially reverses the actions of the original function. If \( f \) takes you from point A to point B, then \( f^{-1} \) takes you from point B back to point A. To find the inverse function’s domain and range, you flip the domain and range of the original function.

• If the original function \( f \) has a range of \([5, \infty)\), then the domain of \( f^{-1} \) is \([5, \infty)\).
• If \( f \) has a domain of \([0, \infty)\), then the range of \( f^{-1} \) is \([0, \infty)\).

This flipping gives us a way to quickly find out what the inverse function will do. It’s like reading a story backward to see how it started.
In our example, the domain and range of the inverse function \( f^{-1} \) are therefore:

1. Domain of \( f^{-1} \) : \([5, \infty)\)
2. Range of \( f^{-1} \) : \([0, \infty)\)

Knowing these makes solving problems about inverse functions much simpler.

Function Properties

Functions have distinct properties that help define their behavior and the nature of their relationships. Here are a few key properties:

• One-to-One Function: Each input maps to a unique output. This is necessary for an inverse function to exist.
• Onto Function: Every element in the range has a corresponding element in the domain.
• Continuous and Discrete: Continuous functions have smooth curves, while discrete functions consist of isolated points.

Understanding these properties can help you intuitively grasp how functions interact with each other and what their inverses might look like.
For example, since our function \( f \) is one-to-one, we know it has an inverse. If it wasn’t one-to-one, there wouldn’t be a unique way to go back from output to input, making the inverse function meaningless. It is these properties that often provide deeper insights and shortcuts for solving function-related problems.