Problem 54 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 53-60: Find a composite function form for \(y\). $$ y=\sqrt[4]{x^{4}-16} $$

Step-by-Step Solution

Answer

The composite function is \( y = v(u(x)) \), with \( u(x) = x^4 - 16 \) and \( v(u) = \sqrt[4]{u} \).

Step 1: Identify the Inner Function

Look inside the given expression and identify the inner function. Here, the expression inside the fourth root is the inner function. Thus, the inner function is: \( u(x) = x^4 - 16 \).

Step 2: Define the Outer Function

Now we identify the outer function, which operates on the result of the inner function. In our case, the outer function is the fourth root, which can be expressed as: \( v(u) = \sqrt[4]{u} \).

Step 3: Compose the Functions

To form a composite function, we substitute the inner function into the outer function. Thus, the composite function \( y \) can be written as: \( y = v(u(x)) = \sqrt[4]{x^4 - 16} \).

Step 4: Write the Composite Function Form

We now explicitly write the composite function using the functions identified: \( y = v(u(x)) \), where \( u(x) = x^4 - 16 \) and \( v(u) = \sqrt[4]{u} \). This completes our definition of the composite function form.

Key Concepts

Inner Function

In understanding composite functions, the inner function forms the crucial foundation that everything else builds upon. The inner function is typically nested within another operation or function. For the exercise at hand, where we have the expression \(y = \sqrt[4]{x^{4} - 16}\), the inner function corresponds to the mathematical operation that is being applied first.
Here, the inner function is the polynomial inside the root, specifically:

• \(u(x) = x^4 - 16\)

This function \(u(x)\) is called the inner function because its calculation must be completed before applying the subsequent ones. Ensuring an accurate understanding of the inner function is vital since it feeds into the next layer of operations.
By breaking down this understanding step-by-step, you can see how each function serves as a building block in the grander scheme of complex operations.

Outer Function

After identifying the inner function, focus shifts to the outer function, which takes the inner function's result as its input. The outer function applies its operation on what the inner function already computed. In our specific example, it involves the action of extracting the fourth root.
We can express the outer function mathematically as:

• \(v(u) = \sqrt[4]{u}\)

This outer function represents a layer of transformation applied to the outcome of the inner function.
Understanding the outer function is key because it dictates the overall transformation that finalizes the expression, modifying the output from the inner function into the desired result. In composite functions, different operations like these combine in a stepwise manner to create complex relationships.

Function Composition

Function composition is the process of executing two or more functions in a specified sequence, utilizing the output from one as input to the next.
For composite functions, this involves nesting functions within each other, creating a chain of operations.

• The composition of the inner and outer functions in our example yields \( y = v(u(x)) = \sqrt[4]{x^4 - 16}\).

The process of composing functions aligns directly with ordered operations, where certain functions must be evaluated before others. To find a composite form, you first solve the inner expression, plug its output into the outer function, and proceed sequentially from there.
This ordered process ensures a systematic transformation of inputs to arrive at the desired final expression. Through practice and application of these concepts, understanding of composite function structure enhances, facilitating more complex mathematical manipulations.