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Chapter 4: Problem 59

Use the graph of the function to determine whether the function is even, odd, or neither. Verify your answer algebraically. $$g(x)=\cot x$$

Short Answer

Expert verified
The function \(g(x) = \cot x\) is neither even nor odd function.

Step by step solution

01

Understand the Graph of the Function

The function given is \(g(x) = \cot x\). The graph of \(\cot x\) is a periodic function which repeats every \(\pi\). More specifically, the cotangent function is undefined at every integer multiple of \(\pi\) and it repeats its values in between these points with a \(-\infty\) to \(\infty\) range.
02

Define Even and Odd Functions

An even function is a function \(f(x)\) that satisfies \(f(-x) = f(x)\) for every number \(x\) in the function's domain. The graph of an even function is symmetric with respect to the y-axis. An odd function is a function \(f(x)\) that satisfies \(f(-x) = -f(x)\) for every number \(x\) in the function's domain. The graph of an odd function is symmetric with respect to the origin.
03

Determine if the function is even, odd, or neither based on the graph

Looking at the graph of \( g(x) = \cot x \), it is noticeable that the function is not symmetric with respect to the y-axis, hence it's not even. Also, the function is not symmetric with respect to the origin, hence it's not odd.
04

Algebraic Verification

Now, we will verify algebraically whether the function is even or odd. If we substitute \(-x\) into the function, we get \(g(-x) = \cot(-x)\). The cotangent function is a reciprocal function to tangent, and the tangent function is odd, hence the cotangent of negative angle is equal to the negative cotangent of the angle. So, \(g(-x) = \cot(-x) = -\cot x = -g(x)\), which confirms that the function \(g(x) = \cot x \) is neither even nor odd.

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