Question

In Exercises \(5-8\), find the inverse function informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=x-5\)

Short answer

The inverse function is \(f^{-1}(x)=x+5\). The relationships \(f(f^{-1}(x))=x\) and \(f^{-1}(f(x))=x\) have been verified to be true.

Step-by-step solution

Find the Inverse Function

To find the inverse function, exchange the roles of \(x\) and \(y\) in the given equation, then solve for the new \(y\). So, if \(f(x)=x-5\), then \(x=y-5\). Solving for \(y\) gives \(y=x+5\). Therefore, the inverse function is \(f^{-1}(x)=x+5\)

Verify \(f(f^{-1}(x))=x\)

Substitute \(f^{-1}(x)\) into the function \(f(x)\) and simplify. Let's substitute \(x+5\) for \(x\) in \(f(x)=x-5\). We get \(f(f^{-1}(x))= (x+5)-5 =x\). Therefore, \(f(f^{-1}(x))=x\) is verified.

Verify \(f^{-1}(f(x))=x\)

Substitute \(f(x)\) into the inverse function \(f^{-1}(x)\) and simplify. Let's substitute \(x-5\) for \(x\) in \(f^{-1}(x)=x+5\). We get \(f^{-1}(f(x))= (x-5)+5 =x\). Therefore, \(f^{-1}(f(x))=x\) is verified.

Key concepts

Algebraic Functions

Algebraic functions are the foundation of many concepts in mathematics and play a crucial role in various fields such as engineering, economics, and the physical sciences. Simply put, an algebraic function is a type of function that involves only algebraic operations – addition, subtraction, multiplication, division, and taking roots. In other words, if you can write the function using a finite number of these operations and it involves only variables and constants, it's an algebraic function.

For example, the function in our exercise, f(x) = x - 5, is an algebraic function because it involves the subtraction of a constant number from the variable x. Such functions are often first introduced in algebra and pre-calculus courses, and they serve as a starting point for students to understand more complex functions and calculus concepts. Understanding how to work with these basic functions, including finding inverses, is essential for tackling more advanced problems.

Function Verification

Function verification is a method used to confirm the correctness of a mathematical function or its related operations. This is especially important when dealing with functions and their inverses. When we talk about verifying a function, we want to show that when we input a value into our function, it produces the expected result.

In our exercise, verification revolves around showing two properties: f(f⁻¹(x)) = x, which confirms that the function f takes the output of the inverse function and returns the original input, and f⁻¹(f(x)) = x, which confirms the inverse function f⁻¹ takes the output of the function and returns the original input.

Application of Verification

Applying verification in our exercise provided peace of mind that the inverse was correctly found. It is a simple, yet powerful tool. By plugging in the inverse back into the original function and vice versa, and ending up with the input x, we have successfully verified both the function and its inverse.

Inverse Function Properties

An inverse function essentially reverses the action of the original function. For every function, ideally, there is an inverse function that can retrieve the original input from its output. However, not all functions have inverses; only bijective functions (ones that are both injective and surjective) do.

The properties of inverse functions are particularly intriguing; they reflect a deep symmetry in mathematics. Here are a few key properties:

• f(f⁻¹(x)) = x and f⁻¹(f(x)) = x when f is bijective.
• The graph of a function and its inverse are reflections of each other across the line y = x.
• If a function f is increasing (or decreasing), so is its inverse.

These properties allow us to understand and predict the behavior of inverse functions, which is critical for solving equations, modeling events, or analyzing relationships in data. In our exercise, recognizing these properties provided certainty that the inverse function f⁻¹(x) = x + 5 was not only correct but functioned as a perfect mirror to the original function through reflection over the identity line.