Question

The function \(f\) is one-to-one. (a) Find its inverse function \(f^{-1}\) and check your answer. (b) Find the domain and the range of \(f\) and \(f^{-1}\). (c) Graph \(f, f^{-1},\) and \(y=x\) on the same coordinate axes. $$ f(x)=\frac{4}{x} $$

Short answer

The inverse function is \(f^{-1}(x) = \frac{4}{x}\). Both functions have a domain and range of \(x eq 0\) or \(( -\infty, 0 ) \cup ( 0, \infty )\). The graphs of \(f(x)\) and \(f^{-1}(x)\) are identical hyperbolas, symmetric about \(y = x\).

Step-by-step solution

Find the Inverse Function

To find the inverse function of a one-to-one function, start by switching the roles of the dependent and independent variables. Set up the equation by letting \(y = f(x)\). Thus, \(y = \frac{4}{x}\). Now, solve for \(x\) in terms of \(y\). Multiply both sides by \(x\) to get \(xy = 4\). Next, divide both sides by \(y\) to isolate \(x\), yielding \(x = \frac{4}{y}\). Finally, swap \(x\) and \(y\) to get the inverse function: \(f^{-1}(x) = \frac{4}{x}\).

Verify the Inverse Function

To verify that \(f^{-1}(x)\) is indeed the inverse of \(f(x)\), the compositions \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\) must both equal \(x\). Check: \(f(f^{-1}(x)) = f\left(\frac{4}{x}\right) = \frac{4}{\left(\frac{4}{x}\right)} = x\). Now check: \(f^{-1}(f(x)) = f^{-1}\left(\frac{4}{x}\right) = \frac{4}{\left(\frac{4}{x}\right)} = x\). Both compositions return the input \(x\), confirming that \(f^{-1}(x) = \frac{4}{x}\) is the correct inverse.

Determine the Domain and Range

Determine the domain and range of \(f(x) = \frac{4}{x}\):1. Domain of \(f(x)\): \(x\) cannot be zero (division by zero is undefined), so the domain is \(x eq 0\) or \(( -\infty, 0 ) \cup ( 0, \infty )\).2. Range of \(f(x)\): Since the function outputs all real numbers except zero, the range is \(y eq 0\) or \(( -\infty, 0 ) \cup ( 0, \infty )\).Next, determine the domain and range of \(f^{-1}(x)\):1. Domain of \(f^{-1}(x)\): By definition, it matches the range of \(f(x)\), so it is \(x eq 0\) or \(( -\infty, 0 ) \cup ( 0, \infty )\).2. Range of \(f^{-1}(x)\): By definition, it matches the domain of \(f(x)\), so it is \(y eq 0\) or \(( -\infty, 0 ) \cup ( 0, \infty )\).

Graph Functions and Line

To graph \(f(x) = \frac{4}{x}\) and \(f^{-1}(x) = \frac{4}{x}\) along with the line \(y = x\) on the same coordinate plane:1. Plot the hyperbola \(f(x)\): This will consist of two branches—one in the first quadrant (positive \(x\) and \(y\)) and one in the third quadrant (negative \(x\) and \(y\)).2. Plot the hyperbola \(f^{-1}(x)\): Since \(f^{-1}(x)\) and \(f(x)\) are the same, this graph coincides with the graph from step 1.3. Plot the line \(y = x\): This line is a 45-degree (diagonal) line passing through the origin, representing where the coordinates of \(f(x)\) and \(f^{-1}(x)\) are symmetric.

Key concepts

One-to-One Function

A one-to-one function is a type of function where each input has a unique output. This means that no two different inputs can produce the same output. Identifying whether a function is one-to-one is crucial when finding its inverse.

To check if a function is one-to-one, you can use the following methods:

• The Horizontal Line Test: If any horizontal line cuts the graph of the function more than once, then it is not one-to-one.
• Algebraic Method: Solve the equation for two different inputs (e.g., if \(f(a) = f(b)\) implies \(a = b\), the function is one-to-one).

For example, with the function \(f(x) = \frac{4}{x}\), a horizontal line will intersect the graph only once, confirming it's a one-to-one function.

Domain and Range

The domain and range of a function are essential concepts that describe the set of possible inputs and outputs, respectively.

For \(f(x) = \frac{4}{x}\):

• Domain: The domain includes all real numbers except zero, as the function is undefined at \(x = 0\). Thus, the domain is \(( -\infty, 0) \cup (0, \infty)\).
• Range: Similar to the domain, the range includes all real numbers except zero, so the range is \(( -\infty, 0) \cup (0, \infty)\).

When considering the inverse function \(f^{-1}(x) = \frac{4}{x}\), roles of domain and range interchange:

• Domain of \(f^{-1}(x)\) matches the range of \(f(x)\): \(( -\infty, 0) \cup (0, \infty)\).
• Range of \(f^{-1}(x)\) matches the domain of \(f(x)\): \(( -\infty, 0) \cup (0, \infty)\).

Graphing

Graphing functions and their inverses helps visualize their behavior and the relationship between them.

To graph \(f(x) = \frac{4}{x}\) and its inverse \(f^{-1}(x)\) (which, in this case, is the same function):

• Plot \(f(x)\): It appears as a hyperbola with two branches—one in the first quadrant and the other in the third quadrant.
• Plot \(f^{-1}(x)\): Since it is the same as \(f(x)\), the graph will overlap with the plot of \(f(x)\).
• Plot the line \(y = x\): This line acts as a mirror. The functions \(f(x)\) and \(f^{-1}(x)\) are symmetric concerning this line.

This visual aid helps understand how the function and its inverse reflect across the line \(y = x\).