Question

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

Short answer

Domain of \(g^{-1}\): \([0, \text{infty})\), Range of \(g^{-1}\): \((-\text{infty}, 0]\).

Step-by-step solution

Understand the given function

Given the function is a one-to-one function \(g\) with domain \((-\text{infty}, 0]\) and range \([0, \text{infty})\). We need to find the domain and range of the inverse function \(g^{-1}\).

Recall properties of inverse functions

For any one-to-one function \(g\), the domain of the inverse function \(g^{-1}\) is the range of \(g\), and the range of \(g^{-1}\) is the domain of \(g\).

Determine the domain of the inverse function

Since the range of \(g\) is \([0, \text{infty})\), the domain of \(g^{-1}\) is \([0, \text{infty})\).

Determine the range of the inverse function

Since the domain of \(g\) is \((-\text{infty}, 0]\), the range of \(g^{-1}\) is \((-\text{infty}, 0]\).

Key concepts

Domain and Range

In mathematics, the domain of a function refers to all the possible input values that the function can accept.
The range, on the other hand, refers to all the possible output values that the function can produce.
Understanding these two concepts is crucial when working with functions and their inverses.
For the given function, which is a one-to-one function noted as \(g\), the domain is \((-\text{∞}, 0]\) and the range is \[0, ∞)\].
When it comes to inverse functions, the domain and range swap roles.
This means that the domain of the inverse function \(g^{-1}\) will be the range of \(g\), and the range of the inverse \(g^{-1}\) will be the domain of \(g\).

One-to-One Functions

A one-to-one function, also known as an injective function, is a function in which every element of the range is mapped by a unique element of the domain.
This means that no two different inputs will map to the same output.
For a function to have an inverse, it must be one-to-one.
In other words, if \(g(x) = g(y)\) implies that \(x = y\), then \(g\) is one-to-one.
A visual way to test if a function is one-to-one is by using the Horizontal Line Test: If any horizontal line intersects the graph of the function at most once, then the function is one-to-one.
Since the function \(g\) in our problem is one-to-one with a given domain and range, we can confidently find its inverse function \(g^{-1}\).

Properties of Inverse Functions

The inverse function of a one-to-one function \(g\) is a function that 'reverses' the mapping of \(g\).
For a function \(g\) and its inverse \(g^{-1}\):

• The domain of \(g\) becomes the range of \(g^{-1}\)
  
• The range of \(g\) becomes the domain of \(g^{-1}\)
  
• Applying \(g^{-1}\) after \(g\) (or vice versa) returns the original input, i.e., \(g(g^{-1}(x)) = x\) and \(g^{-1}(g(y)) = y\)
  

In our problem, the given function \(g\) has a domain of \((-\text{∞}, 0]\) and a range of \[0, ∞)\].
Thus, the inverse function \(g^{-1}\) will have:

• Domain: \[0, ∞)\]
  
• Range: \((-\text{∞}, 0]\)
  

Understanding these properties ensures correct calculations and confirms that we know how to move between a function and its inverse.