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Chapter 0: Problem 30

a. Plot the graph of \(f(x)=\cos (\sin x)\). Is \(f\) odd or even? b. Verify your answer to part (a) analytically.

Short Answer

Expert verified
The graph of the function \(f(x) = \cos(\sin x)\) is symmetric with respect to the \(y\)-axis, indicating that it is an even function. This is analytically verified by showing that \(f(-x) = f(x)\) for the given function.

Step by step solution

01

Plot the graph of the function \(f(x) = \cos(\sin x)\)

To plot the graph of the function \(f(x) = \cos(\sin x)\), we can either use a graphing calculator or an online graphing tool, like Desmos. By using one of these tools, we'll see the graph, which will help us determine its symmetry.
02

Determine if the function is even or odd by checking its symmetry

An even function has the property that \(f(x) = f(-x)\) for all \(x\), while an odd function has the property that \(f(-x) = -f(x)\) for all \(x\). Looking at the graph of \(f(x) = \cos(\sin x)\), you can observe that the function is symmetric with respect to the \(y\)-axis, which indicates that it is an even function.
03

Verify the result analytically

To verify if the function is even or odd analytically, we need to examine \(f(x)\) and \(f(-x)\) for the given function. Let's calculate \(f(-x)\): \(f(-x) = \cos(\sin (-x)) \) Since \(\sin\) is an odd function, we have: \(\cos(\sin (-x)) = \cos(-\sin x)\) Now, since \(\cos\) is an even function, it follows that: \(\cos(-\sin x) = \cos(\sin x)\) Therefore, we can conclude that: \(f(-x) = f(x)\) Thus, the function \(f(x) = \cos(\sin x)\) is even.

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Most popular questions from this chapter

Let \(f\) be a function defined by \(f(x)=\sqrt{x}+\sin x\) on the interval \([0,2 \pi]\). a. Find an even function \(g\) defined on the interval \([-2 \pi, 2 \pi]\) such that \(g(x)=f(x)\) for all \(x\) in \([0,2 \pi]\). b. Find an odd function \(h\) defined on the interval \([-2 \pi, 2 \pi]\) such that \(h(x)=f(x)\) for all \(x\) in \([0,2 \pi]\).

Find the zero(s) of the function f to five decimal places. $$ f(x)=x^{2}-2 x-2 \sin x+1 $$

Find the inverse of \(f .\) Then sketch the graphs of \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=x^{3 / 5}+1 $$

Suppose that \(f\) is a one-to-one function such that \(f(3)=7\) Find \(f\left[f^{-1}(7)\right]\).

Prove that if \(f\) has an inverse, then \(\left(f^{-1}\right)^{-1}=f\).
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