Question

The function \(f(x)=|x|\) is not one-to-one. Find a suitable restriction on the domain of \(f\) so that the new function that results is one-to-one. Then find the inverse of the new function.

Short answer

Restrict the domain to \(x \geq 0\). The new function is \(f(x) = x\), and its inverse is \(f^{-1}(x) = x\).

Step-by-step solution

- Analyze the Original Function

The function given is \(f(x) = |x|\). This is the absolute value function, which is not one-to-one because both positive and negative values of \(x\) yield the same output.

- Identify the Restriction

To make \(f(x)\) one-to-one, restrict the domain of the function to only non-negative or non-positive values. Let's choose the restriction \(x \geq 0\). This means we only consider the right half of the function graph, where the function is increasing and one-to-one.

- Write the New Function

Given the restriction \(x \geq 0\), the new function is simply \(f(x) = x\) because for non-negative values of \(x\), \(|x| = x\). This function is one-to-one.

- Find the Inverse Function

To find the inverse of the new function, swap \(x\) and \(y\) and solve for \(y\). Start with \(y = f(x) = x\). By swapping \(x\) and \(y\), you get \(x = y\). Therefore, the inverse function \(f^{-1}(x) = x\).

Key concepts

Absolute Value Function

An absolute value function is written as \(f(x) = |x|\). This function returns the non-negative value of \(x\), regardless of whether \(x\) is positive or negative. For example, \(f(-3) = | -3 | = 3\) and \(f(3) = | 3 | = 3\). Absolute value functions are often used to measure distance because distance is always non-negative.

The graph of the absolute value function is a V shape and it is symmetrical about the y-axis. This means that for any positive x-value, there is a corresponding negative x-value that results in the same function value. Because of this symmetry, the absolute value function is not one-to-one.

Domain Restriction

To make a function one-to-one, we may need to restrict its domain. A function is one-to-one if every y-value has exactly one corresponding x-value. Since the absolute value function is not one-to-one over all real numbers, we need to limit its domain.

In the case of \(f(x) = |x|\), restricting the domain means choosing either non-negative or non-positive values. By doing so, we ensure that each output comes from only one input. For example, the restriction \(x \geq 0\) means we only consider the part of the graph where the function is always increasing. This makes the function one-to-one.

One-to-One Function

A one-to-one function is a function where every output is associated with exactly one input. In other words, no two different inputs produce the same output. This property is significant because it allows the creation of an inverse function.

For instance, after restricting the domain of the absolute value function to non-negative values, the function \(f(x) = x\) is obtained. This function is one-to-one because each possible output \(y\) corresponds to exactly one value of \(x\). When \(x = y\), every \(y\) from the new domain has a unique pre-image.

Inverse Function Calculation

To find the inverse of a function, we need to swap the roles of \(x\) and \(y\) and then solve for \(y\). This process effectively reverses the function, changing outputs back into their corresponding inputs.

For the modified function \(f(x) = x\) with domain \(x \geq 0\), finding the inverse means starting with \(y = f(x) = x\). We swap \(x\) and \(y\) to get \(x = y\). Solving for \(y\) in this equation, we get \(y = x\). Thus, the inverse function is \(f^{-1}(x) = x\).

The process of finding an inverse shows how the functions directly map each input to a unique output and vice versa. Using inverses is critical in many areas of mathematics and real-life applications where reversing operations is required.