Problem 92 - Precalculus Student Solutions Manual 5th

Question

Answer each of the following For a one-to-one function \(f,\) find \(\left(f^{-1} \circ f\right)(2),\) where \(f(2)=3\)

Step-by-Step Solution

Answer

(f^{-1} \circ f)(2) = 2

Step 1: Understand the meaning of composite functions

The notation \(f^{-1} \circ f\) represents a composite function where \(f^{-1}\) is the inverse function of \(f\). For any \(x\) in the domain of \(f\), \(f^{-1}(f(x)) = x\).

Step 2: Apply the known value in the function

Given that \(f(2) = 3\), we need to find \((f^{-1} \circ f)(2)\). This means we first find \(f(2)\), which is provided as 3.

Step 3: Use the property of inverse functions

By definition of the inverse function \( f^{-1} \circ f \), we have \(f^{-1}(f(2)) = 2\) because \(f^{-1}\) undoes the action of \(f\). Therefore, \((f^{-1} \circ f)(2) = 2\).

Key Concepts

Composite Functions

Composite functions combine two or more functions to form a new function. In the notation \(f \circ g \), this means performing function \(g\) first and then applying function \(f\) to the result. For example, if \(f(x) = x + 1\) and \(g(x) = 2x\), then \(f(g(x)) = f(2x) = 2x + 1\). When working with composite functions, it’s important to follow the order of operations carefully. This helps ensure that all functions are applied correctly. Composite functions are useful for solving more complex problems where the output of one function becomes the input for another.

Inverse Functions

Inverse functions reverse the effect of the original function. If you have a function \(f\) and its inverse \(f^{-1}\), then \(f^{-1}(f(x)) = x\) for all \(x\) in the domain of \(f\). To find an inverse function, solve the equation \(y = f(x)\) for \(x\) in terms of \(y\). This new equation \(x = f^{-1}(y)\) expresses the inverse function. Inverse functions are particularly useful because they allow you to 'undo' the operations of the original function. In the given exercise, we used the property of inverse functions to solve for \((f^{-1} \circ f)(2) = 2\), illustrating how the inverse function undoes the operation of the original function.

Function Properties

Functions have various properties that help us understand their behavior and solve problems. Key properties include:

• One-to-One Functions: A function \(f\) is one-to-one if different inputs produce different outputs. This means \(f(x1) \eq f(x2)\) whenever \(x1 \eq x2\).
• Domain and Range: The domain of a function is the set of all possible inputs, while the range is the set of all possible outputs.
• Injective, Surjective, Bijective: A function is injective if it’s one-to-one, surjective if every element in the range has a pre-image, and bijective if it’s both injective and surjective. For one-to-one functions, finding the inverse is easier because each output corresponds to exactly one input.

These properties play a crucial role in understanding the behavior of functions and solving complex mathematical problems.