Problem 16 - Precalculus : Building Concepts and Connections

Question

Verify that the given functions are inverses of each other. $$f(x)=x^{3}-4 ; g(x)=\sqrt[3]{x+4}$$

Step-by-Step Solution

Answer

The functions f(x) = \(x^{3} - 4\) and g(x) = \(\sqrt[3]{x + 4}\) are inverses of each other.

Step 1: Compute f(g(x))

Substitute g(x) into f(x). That means where you see x in f(x), you replace with g(x): \[f(g(x)) = (g(x))^{3} - 4 = (\sqrt[3]{x+4})^{3} - 4\]

Step 2: Simplify

Simplify the above expression: \[f(g(x)) = x+4 - 4 = x\]

Step 3: Compute g(f(x))

Substitute f(x) into g(x). That means where you see x in g(x), you replace with f(x): \[g(f(x)) = \sqrt[3]{f(x)+4} = \sqrt[3]{(x^{3}-4)+4}\]

Step 4: Simplify

Simplify the above expression: \[g(f(x)) = \sqrt[3]{x^{3}} = x\]

Step 5: Conclusion

Since both f(g(x)) and g(f(x)) resulted to x, therefore, f(x) and g(x) are inverse of each other.

Key Concepts

Verifying Inverse Functions

Understanding the relationship between a function and its inverse is crucial in precalculus, as it gives insight into how two functions can 'undo' each other's actions. To verify that two functions, such as \( f(x) = x^3 - 4 \) and \( g(x) = \sqrt[3]{x+4} \), are indeed inverses, we must show that applying one function followed by the other returns the original input. We accomplish this by performing function compositions: \( f(g(x)) \) and \( g(f(x)) \).

For \( f(g(x)) \), we replace each occurrence of \( x \) in \( f(x) \) with \( g(x) \), and for \( g(f(x)) \), we substitute \( f(x) \) into \( g(x) \). If both compositions simplify to the identity function, which means that \( f(g(x)) = x \) and \( g(f(x)) = x \), then \( f \) and \( g \) are confirmed as inverse functions of each other. The process shown in the solution provides a clear demonstration that both compositions indeed return the original value of \( x \), thus verifying the inverse relationship.

Function Composition

Function composition is a foundational concept in precalculus that allows us to combine two or more functions into a single new function. The composition of two functions, \( f \) and \( g \), is denoted by \( f(g(x)) \), and it represents the output of \( f \) after \( g \) has been applied to an input \( x \). Essentially, it's the process of feeding the output of one function directly into another function.

Understanding how to compute function compositions is key to mastering operations with functions, including the verification of inverse functions. It's important to pay close attention to the order of operations - where you should first apply the innermost function - and to simplify the expression thoroughly, as seen in the provided step-by-step solution.

Cube Root Function

The cube root function is a specific type of radical function that is the inverse of the cubic function. For any real number \( a \), the cube root of \( a \), denoted as \( \sqrt[3]{a} \), is a number that, when raised to the third power, gives \( a \). That is, if \( b = \sqrt[3]{a} \), then \( b^3 = a \).

This function plays a critical role in solving equations involving cubes and in finding the inverse of cubic functions, as seen in our original exercise. When working with cube root functions, it's essential to remember that they undo the action of cubing a number, essentially 'resetting' the input after being processed by a cubic function. As such, considering the cube root function in the exercise, \( g(f(x)) = \sqrt[3]{x^3} \), simplifies nicely to just \( x \), confirming it as an inverse of the cubic function provided.