Problem 17 - College Algebra

Question

For the following exercises, use function composition to verify that \(f(x)\) and \(g(x)\) are inverse functions. $$ f(x)=\sqrt[3]{x-1} \text { and } g(x)=x^{3}+1 $$

Step-by-Step Solution

Answer

Yes, the functions are inverses since both \( f(g(x)) = x \) and \( g(f(x)) = x \).

Step 1: Understand the Concept of Inverse Functions

Two functions \( f(x) \) and \( g(x) \) are inverses of each other if the composition of each function with the other returns the input value \( x \). This means \( f(g(x)) = x \) and \( g(f(x)) = x \).

Step 2: Find \( f(g(x)) \)

Compute \( f(g(x)) \) using the formula for \( f(x) \). Substituting \( g(x) = x^3 + 1 \) into \( f(x) = \sqrt[3]{x - 1} \), we get:\[ f(g(x)) = \sqrt[3]{(x^3 + 1) - 1} = \sqrt[3]{x^3} = x. \]

Step 3: Find \( g(f(x)) \)

Next, compute \( g(f(x)) \) using the formula for \( g(x) \). Substituting \( f(x) = \sqrt[3]{x - 1} \) into \( g(x) = x^3 + 1 \), we get:\[ g(f(x)) = (\sqrt[3]{x - 1})^3 + 1 = x - 1 + 1 = x. \]

Step 4: Verify the Results

Both function compositions \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true for any \( x \). Thus, the functions \( f(x) = \sqrt[3]{x-1} \) and \( g(x) = x^3+1 \) are indeed inverse functions.

Key Concepts

Function Composition

When working with functions, composition is a crucial concept you need to understand. Function composition involves combining two functions to form a new function. If you have functions, say \( f(x) \) and \( g(x) \), the composition of these two functions is written as \( f(g(x)) \). This means you substitute \( g(x) \) into \( f(x) \).
Function composition is used to verify if two functions are inverses. When you compose an inverse function pair, they should "undo" each other. That means both \( f(g(x)) \) and \( g(f(x)) \) will simplify to \( x \).
Understanding this concept can simplify complex problems, especially when considering operations involving various mathematical expressions. It allows you to see how different functions interact and transform inputs.

Cube Root Function

Cube root functions are an interesting type of root function, denoted by \( f(x) = \sqrt[3]{x} \). They deal with the process of finding a number that, when raised to the power of three, results in the original number. For example, \( \sqrt[3]{8} = 2 \), because \( 2^3 = 8 \).
Cube root functions are always defined because any real number has a real cube root. This is significant because it means operations involving cube roots are less restricted compared to square roots that require non-negative radicands.

• Cube Root: The inverse function of the cube function, \( x^3 \).
• Defined for all real numbers.
• Forms an essential part in understanding inverse functions in algebra.

In our exercise, \( f(x) = \sqrt[3]{x-1} \) extends the basic cube root by shifting the input by 1.

Polynomial Function

Polynomial functions are fundamental in algebra and can be classified based on the highest power of the variable present. For example, a cubic polynomial function, \( g(x) = x^3 + 1 \), has its highest term as \( x^3 \).
These functions can have one or more terms, each consisting of a coefficient and a variable raised to a non-negative integer power.

• Cubic Polynomial: Involves terms up to \( x^3 \).
• Can be used in defining invertible functions.
• Essential for modeling many real-world situations.

In the context of the exercise, recognizing \( g(x) = x^3 + 1 \) as a polynomial gives insight into its inverse role with the cube root function, allowing us to see how these operations counteract each other.

Mathematical Verification

Mathematical verification is the process of confirming that a given statement or solution is true. It's a rigorous approach to ensure that the conclusions or results obtained from mathematical operations adhere to established rules and logic.
In our exercise, verification involved checking that the composition of two functions, \( f(g(x)) \) and \( g(f(x)) \), each simplified back to \( x \). This ensures that the functions are genuinely inverses.

• Verifying Results: Ensure that the operations performed on functions accurately meet requirements like simplifying back to the original input.
• Crucial for confirming the accuracy in algebraic functions and their operations.
• Helps build confidence that solutions are correct.

Verification is not just about getting to the end but making sure every step in the solution process aligns with mathematical principles.