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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 \( y = (u^5) \) with \( u = x^4 - 2x^2 + 5 \).

Step by step solution

01

Identify the Inner Function

To break down the given function as a composite function, first identify the inner function. In this case, notice that the expression inside the parentheses is the inner part. Therefore, let \( u = x^4 - 2x^2 + 5 \).
02

Identify the Outer Function

Now, identify the outer function using the inner function. The given expression is raised to the fifth power, so the outer function is the expression being raised to a power. Let \( y = u^5 \).
03

Write the Composite Function Form

With the inner function \( u = x^4 - 2x^2 + 5 \) and the outer function \( y = u^5 \), the composite function form can be written as \( y = (x^4 - 2x^2 + 5)^5 \). This confirms that the form is already in a composite function structure with \( u = x^4 - 2x^2 + 5 \) and \( y = u^5 \).

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Key Concepts

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

Inner Function
When dealing with composite functions, its essential first to identify the inner function. The inner function is located inside another function, which wraps it. This innermost function is like the core, surrounded by an enclosing layer, and can be marked by simple characteristics: usually within parentheses or an argument of trigonometric functions or exponents.
In our original exercise, the inner function is the expression inside the parentheses鈥攕pecifically, the polynomial expression. Here, we have:
  • Inner Function Selection: Choose the part that gets substituted first in other expressions.
  • Mathematical Representation: For this particular problem, \( u = x^4 - 2x^2 + 5 \).
By understanding what constitutes the inner function, we can peel away layers in complex problems, effectively simplifying and making them more manageable.
Outer Function
Once the inner function has been pinpointed, the next step is to examine the outer function. This function operates on the already defined inner function. In simpler terms, the outer function is what you do with the inner function: whether you square it, take the square root, or raise it to another power.
In the exercise, with identified inner part \( u = x^4 - 2x^2 + 5 \), the outer function is what takes this inner part to another realm by applying an operation鈥攊n this case, exponentiation.
Notably, we have:
  • Recognizing the Role of the Outer Function: This is the larger structure or formula affecting the expression's overall outcome.
  • Expression Formation: For our exercise, the outer expression or function is \( y = u^5 \).
Understanding how to distinguish and work with outer functions can vastly enhance your ability to manage complicated algebraic problems with ease.
Function Composition
Function composition is like creating a mathematical sandwich鈥攚here you layer one function within another. The composite function is generally expressed as \( (f \circ g)(x) \), meaning you apply function \( g \) first and then apply function \( f \) to the result of \( g(x) \).
In contexts like the exercise provided, dealing with powerful looking functions becomes approachable by conceptualizing them as multiple simpler functions combined.
The breakdown here allows us to see:
  • Composite Structure: Notice how putting the inner and outer functions together forms the full expression.
  • Application: For the current problem, assembling gives us \( y = (x^4 - 2x^2 + 5)^5 \).
Function composition is not only a crucial topic in calculus and algebra but also an intuitive method to build complex functions from simpler ones, adding significant depth to mathematical understanding.

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Most popular questions from this chapter

(a) Sketch the graph of \(f\) on the given interval \([a, b] .\) (b) Estimate the range of \(f\) on \([a, b] .\) (c) Estimate the intervals on which \(f\) is increasing or is decreasing. $$\begin{aligned} &f(x)=x^{5}-3 x^{2}+1\\\ &[-0.7,1.4] \end{aligned}$$

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Several values of two functions \(T\) and \(S\) are listed in the following tables: $$\begin{array}{|c|ccccc|} \hline t & 0 & 1 & 2 & 3 & 4 \\ \hline T(t) & 2 & 3 & 1 & 0 & 5 \\ \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline S(x) & 1 & 0 & 3 & 2 & 5 \\ \hline \end{array}$$ If possible, find (a) \((T \circ S)(1)\) \((s \circ T)(1)\) (c) \((T \circ T)(1)\) (d) \((S \circ S)(1)\) (e) \((T \circ S)(4)\)

Graph \(f\) in the viewing rectangle \([-12,12]\) by \([-8,8] .\) Use the graph of \(f\) to predict the graph of \(g .\) Verify your prediction by graphing \(g\) in the same viewing rectangle. $$f(x)=x^{3}-5 x ; \quad g(x)=\left|x^{3}-5 x\right|$$

A proposed energy tax \(T\) on gasoline, which would affect the cost of driving a vehicle, is to be computed by multiplying the number \(x\) of gallons of gasoline that you buy by \(125,000\) (the number of BTUs per gallon of gasoline) and then multiplying the total BTUs by the tax \(-34.2\) cents per million BTUs. Find a linear function for \(T\) in terms of \(x .\)
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