Problem 51 - Calculus with Concepts in Calculus

Question

Let \(a\) be a real number and \(f(x)=a-x\). Show that \(f(f(x))=x\) for all \(x\).

Step-by-Step Solution

Answer

For all real numbers \( x \), \( f(f(x)) = x \).

Step 1: Understanding the Function

We are given that the function is defined as \( f(x) = a - x \). Our goal is to find \( f(f(x)) \) and verify if \( f(f(x)) = x \).

Step 2: Applying the Function Iteratively

To find \( f(f(x)) \), first determine \( f(x) \). We have \( f(x) = a - x \).

Step 3: Substitute the Result Back into the Function

Next, use the result from the previous step in the original function again: \( f(f(x)) = f(a-x) \). Substitute \( a - x \) for \( x \) in the function \( f(x) = a - x \).

Step 4: Simplify the Substitution

Now calculate \( f(a-x) \): this becomes \( a - (a - x) \), which simplifies to \( a - a + x = x \).

Step 5: Conclusion

This final expression \( f(f(x)) = x \) confirms that the function is its own inverse for all real numbers \( x \).

Key Concepts

Real Numbers

Real numbers are a fundamental concept in mathematics that include all the numbers on the number line. This encompasses both rational numbers (like fractions and integers) and irrational numbers (such as \( \sqrt{2} \) or \( \pi \)). Understanding real numbers is crucial because they allow us to work with both whole numbers and numbers that have decimal representations.

In the context of the exercise, \( a \) and \( x \) are both real numbers. This means we can perform typical arithmetic operations with them, such as addition and subtraction. These operations are key to solving the equation \( f(f(x)) = x \). By using properties of real numbers, we ensure that our solution is applicable to any real number \( x \). Hence, this shows the broad applicability of our function \( f(x) = a - x \) across the entire set of real numbers.

Substitution in Functions

Substitution is a technique used to simplify complex expressions by replacing variables with other values or expressions. When dealing with functions, substitution allows us to "plug in" values for variables to evaluate or transform the function.

In this exercise, substitution plays a crucial role in finding \( f(f(x)) \). We begin with \( f(x) = a - x \) and then substitute this expression into itself. This means wherever we see \( x \) in \( f(x) \), we replace it with \( a - x \). This results in the new expression \( f(a-x) = a - (a - x) \).

Substituting \( a - x \) into the expression again simplifies to \( x \), demonstrating that \( f(f(x)) = x \), meaning that our function is an inverse of itself. By performing these substitutions carefully, we ensure our solution holds true for all real numbers.

Function Iteration

Function iteration involves applying a function to a result multiple times. In our exercise, we apply the function \( f \) to its result again, a concept known as iterating the function.

To find \( f(f(x)) \), we start with \( f(x) = a - x \). Next, this result is used as an input for the same function, evaluating \( f(a-x) \). The iteration process simplifies to \( f(f(x)) = a - (a - x) \).

Upon simplifying, this double application (or iteration) of \( f \) results in the original input \( x \). This proves that the function is its own inverse, meaning \( f(f(x)) = x \) for any real number \( x \). By understanding function iteration, you can solve similar problems, making this a powerful mathematical concept.