Problem 56 - Algebra and Trigonometry

Question

Express the function in the form \(f \circ g \circ h\) $$ F(x)=\sqrt[3]{\sqrt{x}-1} $$

Step-by-Step Solution

Answer

The function is expressed as \( F(x) = f(g(h(x))) \), where \( h(x) = x \), \( g(x) = \sqrt{x} - 1 \), and \( f(x) = \sqrt[3]{x} \).

Step 1: Identify the Innermost Function

The innermost function in the given expression is the expression inside the square root, which is \( x \). Denote this by \( h(x) = x \). This is the function that first acts on the input \( x \).

Step 2: Identify the Next Function

The next operation on \( x \) is the square root function, but \( \sqrt{x} \) is part of a larger operation in the context of the entire function. Here, the operation performed after \( h(x) \) is finding the square root of \( x \), so we can denote this by \( g(x) = \sqrt{x} \). However, because \( \sqrt{} \) applies directly after, we will slightly adjust: \( g(x) = \sqrt{x} - 1 \). Thus, \( g(h(x)) = \sqrt{x} - 1 \).

Step 3: Identify the Outermost Function

Now, look at the entire expression. After \( x \) is transformed by \( g \), the resulting expression, \( \sqrt{x} - 1 \), is passed into a cube root function. Denote this by \( f(x) = \sqrt[3]{x} \). This function operates on the result of \( g \).

Step 4: Combine the Functions to Form the Composite Function

Combine all the functions into the composite function: \((f \circ g \circ h)(x)\). Therefore, \( f(g(h(x))) = f(\sqrt{x} - 1) = \sqrt[3]{\sqrt{x} - 1} \). This matches the original function: \( F(x) = \sqrt[3]{\sqrt{x} - 1} \).

Key Concepts

Understanding Function Composition

Function composition is a concept where two or more functions are combined to form a new function. This is done by taking the output of one function and using it as the input for the next function.
Think of it like a factory line:

• The first machine takes raw material in.
• The second and third machines use the output of the previous ones to further refine the product.

Function composition follows this idea.In mathematics, this is written as \( (f \circ g)(x) = f(g(x)) \), which means function \( g \) is applied to \( x \, \) and then function \( f \) is applied to the result of \( g(x) \. \) The design is such that complex expressions are simplified systematically.
In our case, we are dealing with three functions, where each one passes its result to the next.

The Outer Function

When we talk about function composition, the outer function is the one that acts last in the sequence. You can think of it as the final step in a multi-step process.
Imagine baking a cake:

• The batter is mixed.
• It is baked.
• Finally, icing is applied.

In this analogy, the icing would be the outer step. For the given problem, the outer function is \( f(x) = \sqrt[3]{x} \. \) This function acts on the result of all the prior processes.
It encapsulates the entire expression and provides the final transformation of \( \sqrt{x} - 1 \) into its cube root. Recognizing the outer function is crucial for understanding how the overall transformation takes place and ends the sequence correctly.

Identifying the Inner Function

Recognizing the inner function is key to simplifying expressions and understanding the function composition process. The inner function operates first, setting the stage for all subsequent operations.
Consider assembling a model airplane:

• You begin by constructing the frame.
• Then you attach additional parts.
• Finally, you paint it.

The frame construction is like an inner function. It’s the foundation.
In our exercise, the innermost function is \( h(x) = x \. \) Here, it simply takes the input \( x \) and passes it forward unaltered.
This role is subtle yet essential, as it initiates the journey through the subsequent functions and leads to the final composition.

Composite Notation Explanation

Composite notation is a neat and concise way of expressing function compositions. It uses the symbol \( \circ \) to indicate the sequence of function operations, much like how you would chain commands in a computer program.
This notation helps convey the idea that multiple functions are acting in a specific order.For example, \( (f \circ g \circ h)(x) \) indicates that \( h \) acts first, followed by \( g, \) and finally \( f. \)
This layered approach offers a clear view of the operation flow, making complex expressions manageable.
By mastering composite notation, you can effectively break down and tackle complex mathematical problems with more confidence and clarity.