Problem 30 - Precalculus with Limits

Question

In Exercises 23-34, show that \(f\) and \(g\) are inverse functions (a) algebraically and (b) graphically. \(f(x) =1 - x^3\), \(g(x) = \sqrt[3]{1 - x}\)

Step-by-Step Solution

Answer

Yes, \(f(x) =1 - x^3\) and \(g(x) = \sqrt[3]{1 - x}\) are inverse functions of each other both algebraically and graphically.

Step 1: Algebraic Proof

First make sure that \(f(g(x)) = x\) and \(g(f(x)) = x\). For \(f(g(x))\) plugging \(g(x)\) into \(f(x)\), gives \(f(g(x)) = 1 - (\sqrt[3]{1 - x})^3 = x\). Similarly for \(g(f(x))\), gives \(g(f(x)) = \sqrt[3]{1 - (1 - x^3)} = x\). Hence, \(f\) and \(g\) are inverse functions algebraically.

Step 2: Graphical Proof

To graph these functions, first graph \(f(x) =1 - x^3\) and then \(g(x) = \sqrt[3]{1 - x}\). It will be seen that these functions are mirror images of each other about the line y=x, which signifies that \(f\) and \(g\) are indeed inverse functions.

Key Concepts

Algebraic Proof of Inverse Functions

Understanding the algebraic proof of inverse functions is essential for grasping how two functions are related inversely. In the provided exercise, we've seen the algebraic steps to demonstrate that the functions f and g are inverses of each other. The calculation involves substituting one function into the other, which should ideally return the original value of x.

The process started with evaluating f(g(x)) and g(f(x)). The former computation involved plugging the expression for g(x) into f(x), which resulted in the identity function, x. This was mirrored in the second computation with g(f(x)), again simplifying to x. These results are consistent with the definition of inverse functions, as applying one function after the other should result in the original input. It's like doing something and then undoing it to end up exactly where you started.

Key Steps in the Algebraic Proof:

• Compute f(g(x)) by substituting g(x) into f(x).
• Perform the necessary algebraic manipulations to simplify the expression.
• Verify that the simplified expression equals x.
• Repeat the process by computing g(f(x)) and simplify.
• If both expressions simplify to x, then the functions are inverses.

Remember, if at any point this process does not result in the identity function, the two functions under consideration are not inverses.

Graphical Proof of Inverse Functions

The concept of inverse functions can also be visualized graphically, which reinforces the algebraic proof. In the given problem, we've established that f(x) = 1 - x^3 and g(x) = \(\sqrt[3]{1 - x}\) are inverse functions. A graphical representation provides a more intuitive understanding of this relationship.

When graphing these functions, we should see that one function is the mirror image of the other about the line y=x. This is because each (x, y) coordinate point on the graph of f has a corresponding (y, x) point on the graph of g. The line y=x acts like a mirror, reflecting any point across to its inverse counterpart.

Steps for Graphical Verification:

• Plot the function f(x) on a coordinate system.
• Plot the function g(x) on the same coordinate system.
• Draw the line y=x to act as a reference for reflection.
• Check for symmetry of the two graphs relative to the line y=x.

If the functions f and g are true inverses, the graphical representation will confirm the reflection symmetry, just as it was detailed in the step-by-step solution.

Composition of Functions

The relation between inverse functions is closely linked with the concept of composition of functions. Composition involves combining two functions to form a new function by applying one function to the result of another. Mathematically, the composition of functions f and g is written as f(g(x)) or g(f(x)).

In the case of inverse functions, the composition of a function and its inverse yields the original value. This means that f(g(x)) equals x, as does g(f(x)). In terms of function operation, it's as if the action of f is undone by g, restoring the starting value.

Understanding Composition:

• The notation f(g(x)) means that g(x) is applied first, and then f is applied to the result.
• The goal with inverse functions is to arrive back at the input, x, after the composition.
• If functions are inverses, their composition in any order will result in the identity function, x.

The composition is an important operation in mathematics as it demonstrates the interaction between functions and reveals properties such as invertibility, which is crucial for functions to be considered inverses.