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Chapter 4: Problem 58

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

Short Answer

Expert verified
The function \(f(x) = \tan x\) is an odd function both graphically and algebraically.

Step by step solution

01

Analyze the graph of the function

By simply looking at the graph of the given function, it becomes clear that the function \(f(x) = \tan x\) is symmetric with respect to the origin.It means the graph is symmetric about the origin, hence the function is odd .
02

Algebraic Verification for Oddness

To verify that \(f(x) = \tan x\) is an odd function algebraically, find \(f(-x)\) and compare it with \(-f(x)\). \n If \(f(-x) = -f(x)\), then the function is odd. Substituting \(-x\) in place of \(x\) in \(f(x)\) gives: \n \(f(-x) = \tan(-x) = -\tan(x)\), because the tangent function is odd, i.e., \(\tan(-x) = -\tan(x)\). \n Thus, \(f(-x) = -f(x)\), confirming that \(f(x) = \tan x\) is an odd function.

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