Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=x $$

Step-by-Step Solution

Answer

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

Step 1: Understand the given function

The function given is \( f(x) = x \). This is a linear function where the output is exactly the same as the input for any value of \( x \).

Step 2: Recall the definition of an inverse function

An inverse function of \( f(x) \), denoted as \( f^{-1}(x) \), undoes the action of \( f(x) \). In other words, \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

Step 3: Assume the form of the inverse

Assume that the inverse function \( f^{-1}(x) \) is such that when you apply \( f \) to it, you get back \( x \). Since \( f(x) = x \), for any \( x \), the inverse function must also be that it outputs \( x \) from itself.

Step 4: Verify the inverse

Check the equations \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). Apply \( f \) to \( f^{-1}(x) \):Since \( f^{-1}(x) = x \), then \( f(f^{-1}(x)) = f(x) = x \).Now, apply \( f^{-1} \) to \( f(x) \):Since \( f(x) = x \), then \( f^{-1}(f(x)) = f^{-1}(x) = x \).

Step 5: Conclude the inverse function

Since \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \), we have verified that the function \( f(x) = x \) is its own inverse. Therefore, the inverse function \( f^{-1}(x) = x \).

Key Concepts

Linear Function

A linear function is a basic algebraic function with the general form of \( f(x) = ax + b \), where \( a \) and \( b \) are constants. However, in the specific case of the function given in the exercise, \( f(x) = x \), which is known as the identity function, \( a = 1 \) and \( b = 0 \).This means that whatever value you input, the output is exactly the same.

Key features of linear functions include:

• The graph is a straight line.
• The slope, \( a \), determines how steep the line is.
• There is no curve, making it one of the simplest functions to work with.

Linear functions serve as a foundation for understanding more complex algebraic functions.They also help in visualizing the idea of slope and intercept which are crucial when dealing with equations of different lines.

Function Properties

Functions have specific properties that make them a useful tool in various mathematical situations. Knowing these properties helps to understand how functions interact with each other, like when determining inverses.

Some fundamental properties of functions are:

• One-to-One: A function is one-to-one if every output is determined by exactly one input. This property is crucial for a function to have an inverse because only one-to-one functions can be uniquely reversed.
• Onto: A function is onto if every possible value in the output set is covered by the function, ensuring that the function can "hit" every number in the desired range.
• Identity: The identity function (like \( f(x) = x \)) has the property where each element is mapped to itself, making it its own inverse.

Identifying these properties makes it easier to determine whether a function can have an inverse and how that inverse would work.

Algebraic Function

An algebraic function is a type of mathematical expression created from integers and variables using basic operations such as addition, subtraction, multiplication, division, and taking roots. Linear functions, polynomials, and rational functions are examples of algebraic functions.

For algebraic functions, certain operations can help with transformations and solving equations:

• Solving for Inverses: Swapping variables and solving allows us to find the inverse of certain algebraic functions. In the given exercise with \( f(x) = x \), there isn't any swapping really needed as the inverse is the same function.
• Simplification: Break down complex expressions into simpler components to ease calculations.

Understanding how algebraic functions operate and transform is essential in solving problems related to inverses, simplifying complex equations, and connecting algebraic concepts to graphical representations.