Problem 57 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 53-60: Find a composite function form for \(y\). $$ y=\left(x^{4}-2 x^{2}+5\right)^{5} $$

Step-by-Step Solution

Answer

The composite function form is \( y = (u(x))^5 \) where \( u(x) = x^4 - 2x^2 + 5 \).

Step 1: Identify the Inner and Outer Functions

To express the given function as a composite function, we first identify the inner and the outer functions. The given function is: \( y = (x^4 - 2x^2 + 5)^5 \). Here, the inner function \( u \) is \( u = x^4 - 2x^2 + 5 \) and the outer function is \( y = u^5 \).

Step 2: Express the Composite Function

Using the identified inner and outer functions, express \( y \) as a composite function. Since the inner function is \( u(x) = x^4 - 2x^2 + 5 \) and the outer function is \( y(u) = u^5 \), the composite function is given by \( y(u(x)) = (u(x))^5 \). Substitute the expression for \( u \) to get \( y = (x^4 - 2x^2 + 5)^5 \), which is already in composite form.

Key Concepts

Inner Function

In the world of mathematics, understanding the concept of the inner function is crucial when dealing with composite functions. The inner function is the function that is applied first in a composition. In the given exercise, we need to identify what function is being "plugged into" another. This function is the one inside the parentheses when you express the composite function.

For the expression \( y = (x^4 - 2x^2 + 5)^5 \), the inner function, often represented by \( u \), is \( u = x^4 - 2x^2 + 5 \).

• This function takes \( x \) as its input.
• It produces an output that is used by another function.

Recognizing the inner function is the first essential step in breaking down a composite function. This function effectively maps \( x \) to a new value \( u \), setting the stage for the outer function to act upon this new input.

Outer Function

The outer function is the second function that acts on the result of the inner function. It is what you apply to the output of the inner function, transforming it again into something new. The outer function takes the output from the inner function as its input.

In the example of \( y = (x^4 - 2x^2 + 5)^5 \), the outer function can be expressed as \( y(u) = u^5 \).

• This function "wraps around" the inner function.
• It essentially elevates the result of \( u = x^4 - 2x^2 + 5 \) to the fifth power.

In composite functions, the outer function typically adds the final transformation that converts the intermediate results from the inner function into the desired solution or final output of the original expression.

Function Composition

Function composition is a fundamental concept in mathematics and can be thought of as applying one function to the results of another. This involves two (or more) functions working together in sequence. When we consider a composite function, it's like a chain reaction where one function's output serves as the input for another.

With our example function, \( y = (x^4 - 2x^2 + 5)^5 \), the composition is expressed as \( y(u(x)) \). Here's what that means:

• The inner function \( u(x) = x^4 - 2x^2 + 5 \) is calculated first.
• The result from \( u(x) \) is then used as input in the outer function \( y(u) = u^5 \).

So the composite function \( y \) is a result of combining these two functions into a single operation, where the process moves seamlessly from calculating \( u(x) \) to applying \( y(u) \).

Understanding function composition is not just about knowing how to mechanically carry out operations; it's about appreciating how different mathematical functions can interact and be combined to model complex situations.