Problem 71 - Calculus

Question

Explain what is wrong with the statement. The inverse of \(f(x)=x\) is \(f^{-1}(x)=1 / x\).

Step-by-Step Solution

Answer

\( f(x) = x \) is its own inverse; \( f^{-1}(x) = 1/x \) is incorrect.

Step 1: Understanding the Original Function

The function given is \( f(x) = x \). This is a linear function where each input \( x \) outputs the same value \( x \). So for every \( x \), \( f(x) = x \).

Step 2: Defining an Inverse Function

An inverse function \( f^{-1} \) is one that, when composed with the original function \( f \), returns the input value: \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

Step 3: Calculating the Inverse of \( f(x) = x \)

To find the inverse, we set \( y = f(x) = x \) and then solve for \( x \). In this case, since \( y = x \), switching \( x \) and \( y \) gives \( x = y \), indicating \( f^{-1}(x) = x \). Thus, the function is its own inverse.

Step 4: Analyzing the Given Inverse \( f^{-1}(x) = \frac{1}{x} \)

The statement claims \( f^{-1}(x) = \frac{1}{x} \), but this would imply \( f(f^{-1}(x)) = f(\frac{1}{x}) \), which is \( \frac{1}{x} \), not \( x \). Therefore, \( \frac{1}{x} \) is not a valid inverse for \( f(x) = x \).

Step 5: Conclusion

The error in the statement is that \( f^{-1}(x) = \frac{1}{x} \) is incorrect. The correct inverse for \( f(x) = x \) is \( f^{-1}(x) = x \), because this function itself is its own inverse.

Key Concepts

Linear Function

A linear function is a simple mathematical operation where each input value is directly proportional to the output. The general form of a linear function can be written as \( f(x) = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept. In our exercise, the function \( f(x) = x \) is a special case of a linear function where \( m = 1 \) and \( b = 0 \).
The function simply maps an input \( x \) to the same output \( x \). Since it is a straight line through the origin with a slope of 1, it has a unique quality: each point on the line is directly reflective of every other.

• Linear functions always produce graphs that are straight lines.
• They have constant slopes.
• In this particular case, \( f(x) = x \), the line has a slope of 1.
• They are defined for all real numbers and do not have restrictions on input values.

Composite Function

Composite functions are created when the output of one function becomes the input to another. If we have two functions, \( f \) and \( g \), their composite is written as \( f(g(x)) \). The essence of a composite is linking operations and observing how they affect each other.
For example, if \( g(x) = 2x \) and \( f(x) = x + 3 \), then the composite \( f(g(x)) \) is \( f(2x) = 2x + 3 \).
Understanding composition is essential when finding inverse functions because it clarifies how functions interact and "undo" one another.

• Composite functions reveal the order in which operations are applied.
• They can highlight dependencies between operations.
• The composition \( f(f^{-1}(x)) = x \) shows how a function and its inverse rearrange to return the original input.
• Thinking of composition simplifies solving for unknowns across linked functions.

Inverse Calculation

Calculating the inverse of a function involves finding a function that "undoes" the effect of the original function.
To determine the inverse, set \( y = f(x) \) and solve for \( x \) in terms of \( y \). Then switch the roles of \( x \) and \( y \), naming the new function \( f^{-1}(x) \).
In our exercise's linear function \( f(x) = x \), we start by setting \( y = x \). Since swapping the variables yields \( x = y \), the inverse function is \( f^{-1}(x) = x \), underscoring how this function is its own inverse.

• Inverting means reversing operations.
• Once identified, an inverse maintains identity in functional compositions.
• The goal is always to return the original input value.
• For linear functions like \( x = f(x) \), the inverse calculation is straightforward.

Function Analysis

Analyzing functions involves inspecting their behavior, graphed results, and relationships with other functions. In the case of inverse functions, a keen analysis uncovers errors, such as when the wrong inverse is identified. The aim is to verify correctness through composition and critical problem-solving.
Our primary error was assuming \( f^{-1}(x) = \frac{1}{x} \) was an inverse to \( f(x) = x \). Proper analysis using composition \( f(f^{-1}(x)) \) quickly identifies such mismatches where inputs are not returned to their original state.

• Graphing functions helps visualize output relationships.
• Analysis requires checking against confirmations, like inverses through composites.
• Errors in assumptions like an inverse are often revealed by unexpected results.
• Ensuring functional operations observe their intended transformations solidifies analyses.