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Chapter 2: Problem 57

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

Short Answer

Expert verified
The composite function is \( y = f(g(x)) = (g(x))^5 \) where \( g(x) = x^4 - 2x^2 + 5 \).

Step by step solution

01

Identify the Outer Function

Start by recognizing the outer function of the given expression, which is raised to the power of 5. Therefore, the outer function can be defined as \( f(u) = u^5 \).
02

Identify the Inner Function

Next, identify the inner function represented by the expression inside the parentheses. Thus, the inner function can be defined as \( g(x) = x^4 - 2x^2 + 5 \).
03

Form the Composite Function

Construct the composite function by substituting the inner function into the outer function. The composite function can be expressed as \( y = f(g(x)) = (g(x))^5 = (x^4 - 2x^2 + 5)^5 \).
04

Confirm the Composite Form

Review the form to ensure that it accurately represents the original function. In its composite form, it reads \( y = f(g(x)) = (g(x))^5 \) which matches the given expression \( (x^4 - 2x^2 + 5)^5 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Outer Function
Understanding the outer function is crucial when working with composite functions. In our given problem, the outer function represents the operation or transformation applied last. It encompasses the complete inner function as its input. With the expression \( y=(x^4-2x^2+5)^5 \), the outer function can be identified by the exponent of 5.

Here's a breakdown to help simplify this concept:
  • In mathematical terms, the outer function is defined as \( f(u) = u^5 \).
  • **Why 'u'?** The 'u' stands for a placeholder variable that will later be replaced by the actual inner function.
  • By raising to the power of 5, the outer function transforms the inner result by squaring its input.
Identifying the outer function allows you to recognize the main transformation happening to the inner function's output. It is like wrapping the core function result in another layer.
Inner Function
The inner function forms the core component of composite functions. It gets evaluated first, becoming the input for the outer function. In our problem, the expression inside the parentheses \( (x^4-2x^2+5) \) serves as the inner function.

Here鈥檚 how we break it down:
  • The inner function is defined as \( g(x) = x^4 - 2x^2 + 5 \).
  • This function is a polynomial with three terms: \( x^4 \), \(-2x^2 \), and \(+5 \).
  • Each component affects the shape and characteristics of the function graph.
Why is it important? The inner function sets the stage for transformation by the outer function. Recognizing it allows you to know what you're preparing for the next layer of computation.
Function Composition
Function Composition is a fundamental concept when dealing with complex functions. It involves plugging one function into another, creating a new, unified function.

Think of composing functions like a two-step process:
  • First, compute the inner function: \( g(x) = x^4 - 2x^2 + 5 \).
  • Then, use that result as the input for the outer function: \( f(u) = u^5 \).
By substituting the inner function into the outer one, we form the composite function. This is expressed as \( y = f(g(x)) \), which reads: "apply \( g(x) \) first, then apply \( f(u) \)."

Here are some tips to remember:
  • Composite functions enable us to perform multiple operations in a structured manner.
  • They allow for more elegant and simplified expressions.
By mastering function composition, you gain the ability to tackle a wide array of mathematical problems with confidence and skill.

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