Problem 10 - College Algebra

Question

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$ f(x)=\sqrt[3]{x-4} \text { and } g(x)=x^{3}+4 $$

Step-by-Step Solution

Answer

The function compositions \(f(g(x))\) and \(g(f(x))\) both equal x. Therefore, \(f\) and \(g\) are inverses of each other.

Step 1: Compute \(f(g(x))\)

To find \(f(g(x))\), you substitute \(g(x)\) into our function \(f(x)\). So, \(f(g(x))= f(x^{3}+4)\). From the definition, we know that \(f(x)= \sqrt[3]{x-4}\). Therefore, upon substituting, \(f(g(x)) = \sqrt[3]{(x^3+4)-4} = \sqrt[3] {x^3}\).

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

Now we compute \(g(f(x))\) similarly. You substitute \(f(x)\) into \(g(x)\) to get \(g(f(x))= g(\sqrt[3]{x-4})\). Since \(g(x) = x^3 + 4\), substituting gives \(g(f(x)) = (\sqrt[3]{x-4})^3 + 4 = x - 4 + 4 = x\).

Step 3: Determine whether \(f\) and \(g\) are inverses

Functions \(f\) and \(g\) are inverses of each other if and only if \(f(g(x)) = x\) and \(g(f(x)) = x\) for all x in the domains of \(g\) and \(f\) respectively. From Steps 1 and 2, we find that \(f(g(x)) = x\) and \(g(f(x)) = x\), meaning that \(f\) and \(g\) are inverses of each other.

Key Concepts

Composite Functions

Understanding composite functions is like solving a puzzle where one piece is hidden inside another. Essentially, a composite function is created when one function is applied to the results of another function. It's represented as \(f(g(x))\) or \(g(f(x))\), where the outcome of function \(g\) becomes the input for function \(f\), or vice versa.

For example, consider a situation where you want to paint a wooden block. First, you shape the block (function \(g\)), and then you paint it (function \(f\)). Similarly, in algebra, first you apply operation \(g(x)\), and then use the result as the input for operation \(f(x)\).

In the exercise provided, \(f(g(x))\) means you first apply \(g(x)\), which is \(x^{3}+4\), and then apply \(f\) on the result, ending up with \(f(x^3+4)\). The process effectively unravels the nested functions, leading to a simple expression. It's like unpacking nested boxes until you find the item you've been searching for.

Inverse Functions

Inverse functions reverse the process of each other; think of them as funhouse mirrors—they reflect back the original value. For a function \(f\) to have an inverse, every input must lead to a unique output, and vice versa. This ensures the original input can be retrieved after applying the function and its inverse in succession.

When functions \(f\) and \(g\) are inverses, the compositions \(f(g(x))\) and \(g(f(x))\) will both simplify down to \(x\). If you go through the exercise, you'll observe this happening. It's like putting on a pair of glasses that makes blurry images sharp: by applying the inverse function, you're able to see the original input clearly again.

This plays a key role in solving equations and understanding transformations in various mathematical contexts. The revelation that functions \(f(x) = \sqrt[3]{x-4}\) and \(g(x) = x^{3}+4\) are indeed inverses supports the idea that they perfectly 'undo' each other.

Function Operations

Just like numbers can be added, subtracted, multiplied, and divided, functions too can undergo similar operations. When you perform an operation on two functions, you create a new function with its own set of properties.

For instance, adding two functions \(f(x) + g(x)\) involves adding the outputs of \(f\) and \(g\) for the same input \(x\). Subtracting, multiplying, and dividing function operations follow the same principle of performing the respective arithmetic operation on the outputs.

In the exercise, we applied operations to nest one function inside the other, creating composite functions, which is another way functions can be combined. Each operation on functions can affect their graphs and properties significantly, which is why understanding these operations is fundamental to mastering algebra.

Algebraic Functions

Algebraic functions, such as the ones in our exercise, are functions that involve basic algebraic operations—addition, subtraction, multiplication, division, and root extraction, often featuring polynomial expressions. The function \(f(x)= \sqrt[3]{x-4}\) involves a cube root, a specific type of root extraction. Meanwhile, \(g(x)=x^{3}+4\) is a polynomial function due to the cube and constant.

These functions describe a wide variety of relationships and can represent real-world phenomena, making them incredibly useful tools in many fields, including science and engineering. They’re the foundation stones of higher mathematics and understanding their behavior is essential for advancing in mathematical studies.

Analytics plays a crucial role, too, as algebraic functions can be graphed to visually display their characteristics, helping us better grasp their nature and how two such functions can interact as inverses.