Problem 52 - Algebra and Trigonometry

Question

\(51-54\) 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 \(F(x)\) is expressed as \(f(g(h(x)))\) with \(h(x) = \sqrt{x}\), \(g(y) = y - 1\), \(f(z) = \sqrt[3]{z}\).

Step 1: Understand the Given Function

The function provided is \(F(x) = \sqrt[3]{\sqrt{x} - 1}\). This means you're taking the square root of \(x\), subtracting 1 from it, and then taking the cube root of the result.

Step 2: Define Function \(h(x)\)

Identify the first operation in the function from the inside out. The innermost operation is taking the square root of \(x\). Therefore, define \(h(x) = \sqrt{x}\).

Step 3: Define Function \(g(y)\)

The next operation involves subtracting 1 from the result of \(h(x)\). Define this as \(g(y) = y - 1\). Here, \(y\) represents the output from \(h(x)\).

Step 4: Define Function \(f(z)\)

The final operation is taking the cube root of the result from \(g(y)\). Define this as \(f(z) = \sqrt[3]{z}\). Here, \(z\) is the output from \(g(y)\).

Step 5: Verify the Composition

Put it all together using the composition of functions, starting from the inside: \(h(x) = \sqrt{x}\), then apply \(g(y) = y - 1\) with \(y = h(x)\), and finally \(f(z) = \sqrt[3]{z}\) with \(z = g(y)\). Thus, \(F(x) = f(g(h(x)))\).

Key Concepts

Function Transformation

Transforming functions involves altering their basic shape, orientation, or position on a graph. When we talk about transformation, it could mean translation (moving the graph up, down, left, or right), dilation (stretching or compressing), reflection (flipping over an axis), or rotation (turning around a point). In the exercise, the function transformation is about expressing a complex function into simpler parts.

• Translation: Moving the graph horizontally or vertically. For example, adding or subtracting a value shifts the graph up or down.
• Dilation: Changing the graph's size. Multiplying by a factor greater than 1 stretches it, and less than 1 compresses it.
• Reflection: Mirror images of the graph about an axis, such as flipping it upside down.
• Rotation: Turning the graph around a certain point.

Recognizing these transformations helps in dissecting and understanding complex mathematical functions like the one given in the exercise, which combines square and cube roots.

Nested Functions

Nested functions are functions placed inside each other. In the context of the exercise, we observe this in the form of a function inside another function — like layers of an onion. The concept is crucial because it affects how we evaluate or modify the overall function.
Consider the expression given:

• Innermost Function: The basic function, or the core, that you start with. Here, it is taking the square root of \(x\).
  
• Secondary Function: The function wrapping around the innermost. It modifies the result of the first function. In our example, it's subtracting 1 from the square root result.
  
• Outermost Function: This is the final layer, affecting the outcomes from all previous functions. In this case, it's taking the cube root.
  

Understanding nested functions helps unravel the order and construction of expressions, simplifying the approach to solving or manipulating complex functions. By breaking down elaborate compositions into smaller, manageable segments, learners gain better insight into function behavior.

Order of Operations

Order of operations is a critical concept in mathematics that dictates the sequence in which calculations should be conducted to ensure consistent results. This principle becomes particularly important when dealing with complex expressions or nested functions as seen in the exercise.
The standard order of operations is encapsulated in the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)). For functions, this translates to performing operations from the innermost to the outermost layers.

• Innermost Operations: Always start with the deepest nested function or parentheses level. Here, it's calculating the square root of \(x\).
• Intermediate Operations: Proceed to the operations that come next. Subtracting 1 follows once the square root has been determined.
• Outermost Operations: Conclude with the outermost operations. Cube root is applied last in the given function transformation.

Ensuring the right order of operations helps avoid mistakes in calculation and guarantees accuracy in function evaluations. For students, mastering this order is foundational for tackling intricate mathematical problems with confidence and precision.