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Chapter 3: Problem 4

Identify the inner and outer functions in the composition \(\cos ^{4} x\)

Short Answer

Expert verified
Answer: The inner function is \(\cos x\) and the outer function is \(x^4\).

Step by step solution

01

Identify the inner function

The inner function is the one that is inside of another function. In this case, the cosine function is inside the power function. So, the inner function is: $$ f(x) = \cos{x} $$
02

Identify the outer function

The outer function is the one that is applied to the results of the inner function. In this case, the power function is applied to the cosine function. So, the outer function is: $$ g(x) = x^4 $$
03

Write the composition of functions

Now that we have identified the inner and outer functions, we can write the composition of functions as: $$ (\cos^4{x}) = g(f(x)) = g(\cos{x}) = (\cos{x})^4 $$ This shows that the inner function is \(\cos x\) and the outer function is \(x^4\).

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Key Concepts

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

Inner Function
In the context of composite functions, the inner function is the function that is "nested" inside another function. It is the initial function that is executed first in a sequence of operations. To put it simply, it's the core function at the heart of your mathematical operation.
  • In our example, the inner function is the cosine function expressed as \( f(x) = \cos{x} \).
  • This means that whatever operation we perform first will be applied to this cosine function.
Imagine you are constructing a sequence of operations. The inner function is "wrapped" by any additional functions applied later, such as a power or logarithm, known as outer functions.
Understanding the inner function helps in breaking down complex expressions into manageable parts.
Outer Function
The outer function in a composition of functions acts like a shell or cover, encompassing the inner function. It is the last function that acts on the result of the inner function. Recognizing the outer function involves identifying what operation is performed last in the expression.
  • In our case, the outer function is the power function given by \( g(x) = x^4 \).
  • This indicates that once the cosine function is resolved, this power function is then applied, raising the entire expression to the fourth power.
Think of the outer function as the ultimate task in a sequence of functions: Multiplicate, exponentiate, etc., it defines the final step in your calculation. It's crucial to identify the outer function because it dictates the final transformation of the inner function's result.
Cosine Function
The cosine function is a fundamental trigonometric function used to model periodic phenomena such as sound waves, light waves, and mechanical vibrations. It is typically represented as \( \cos{x} \), where \( x \) could be an angle in radians or degrees.
  • The cosine function yields values ranging from -1 to 1.
  • It is significant in calculating angles and lengths in right triangles, contributing to both geometry and calculus.
In the composition \( \cos^{4} x \), the cosine function acts as the inner function; it evaluates the angle first, then the outer function raises the outcome to the fourth power. Grasping the cosine function is essential in many areas, because it serves as the building block for more complex trigonometric identities and computations, making it versatile for various calculations.

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Most popular questions from this chapter

Given the function \(f,\) find the slope of the line tangent to the graph of \(f^{-1}\) at the specified point on the graph of \(f^{-1}\) . $$f(x)=(x+2)^{2} ;(36,4)$$

Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}\left[\frac{f(x) g(x)}{x}\right]\right|_{x=4}$$

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Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}(x f(x))\right|_{x=3}$$

Proof of \(\lim _{x \rightarrow 0} \frac{\cos x-1}{x}=0\) Use the trigonometric identity \(\cos ^{2} x+\sin ^{2} x=1\) to prove that \(\lim _{x \rightarrow 0} \frac{\cos x-1}{x}=0 .(\) Hint: Begin by multiplying the numerator and denominator by \(\cos x+1 .)\)
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