Problem 21 - College Algebra Essentials

Question

The functions in Exercises \(11-28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\) $$f(x)=\frac{1}{x}$$

Step-by-Step Solution

Answer

The inverse function of \(f(x)=\frac{1}{x}\) is \(f^{-1}(x) = \frac{1}{x}\)

Step 1: Find the inverse function

To determine the inverse function \(f^{-1}(x)\), we switch \(x\) and \(y\) in the original function and solve for \(y\). In this case, we have: \[x=\frac{1}{y}\] Use cross multiplication: \[xy = 1\] Then solve for \(y\): \[y = \frac{1}{x}\] Therefore, \(f^{-1}(x) = \frac{1}{x}\].

Step 2: Verify the inverse function is correct

We need to confirm that \(f(f^{-1}(x))=x\) and \(f^{-1}(f(x))=x\). For the former, substitute \(f^{-1}(x)\) into \(f(x)\): \[f(f^{-1}(x))=f\left(\frac{1}{x}\right)=x\] For the latter, substitute \(f(x)\) into \(f^{-1}(x)\): \[f^{-1}(f(x))=f^{-1}\left(\frac{1}{x}\right)=x\]

Step 3: Conclude the results

From above steps, we can conclude that both conditions for the inverse function are met (i.e., \(f(f^{-1}(x))=x\) and \(f^{-1}(f(x))=x\)). So, our inverse function \(f^{-1}(x) = \frac{1}{x}\) of \(f(x) = \frac{1}{x}\) is proved correct.

Key Concepts

Understanding One-to-One Functions

A one-to-one function is a special type of function where each input value has a unique output value, and vice versa. This uniqueness is crucial because it ensures that the function has an inverse. Think of it like a pair of shoes - each shoe fits specifically on the left or right foot, just as each input has a distinct output, and no two inputs share the same output.

• To determine if a function is one-to-one, you can use the Horizontal Line Test. If no horizontal line cuts the graph more than once, then the function is one-to-one.
• This characteristic allows each output to be directly traced back to one specific input, which is necessary for the existence of an inverse function.

Understanding one-to-one functions better helps in grasping inverse functions because not every function has an inverse, only these special ones do.

The Process of Function Verification

Function verification is the process of checking whether a proposed inverse function truly behaves as expected.

Verification involves two important checks:

• First, confirm that applying the function to its inverse returns the original input: \( f(f^{-1}(x)) = x \).
• Second, verify that applying the inverse to the function does the same: \( f^{-1}(f(x)) = x \).

These checks ensure the correctness of the inverse. Both must hold true to confidently say the proposed inverse truly functions as the reverse of the original function. If they do not simultaneously succeed, errors may exist in finding or expressing the inverse.

Exploring Inverse Function Properties

Inverse functions "reverse" the effects of the original function. They can be thought of as the function "undoer." When you apply an inverse function to its counterpart function, they cancel each other out, leaving the initial value intact.

Key Properties include:

• Existence: Only one-to-one functions have inverses. (Recall our earlier section on one-to-one functions!)
• Symmetry: The graph of an inverse function is a reflection of the original function across the line \(y = x\). This line acts as a mirror, showcasing symmetry between a function and its inverse.
• Notation: The inverse of a function \(f\) is expressed as \(f^{-1}\). Note that this is not the same as \(\frac{1}{f}\), but rather signifies the function that reverses \(f\).

Understanding these properties is key to mastering how inverse functions operate and interact, making it easier to manipulate and verify them in exercises.