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Chapter 5: Problem 25

Is arctangent an even function, an odd function, or neither?

Short Answer

Expert verified
The arctangent function (tan^(-1)(x) or arctan(x)) is an odd function, as it satisfies the condition \(arctan(-x) = -arctan(x)\).

Step by step solution

01

Define the arctangent function and conditions for even and odd functions

The arctangent function is given by y = tan^(-1)(x) or y = arctan(x). Recall that for a function to be even, it must satisfy f(x) = f(-x), and for a function to be odd, it must satisfy f(-x) = -f(x).
02

Test if arctangent is an even function

To test if arctangent is an even function, we need to check if arctan(x) = arctan(-x). Let's try to find the relationship between these two: arctan(-x) can be rewritten as the angle whose tangent is -x. If we take the tangent of both sides, we get: tan(arctan(-x)) = -x We know that tangent is an odd function, which means tan(-y) = -tan(y). Thus, we can rewrite the left-hand side as: tan(-arctan(x)) = -x This equation tells us that the tangent of the angle -arctan(x) is equal to -x, but it does not imply that arctan(x) = arctan(-x). So, the arctangent function does not satisfy the condition for being even.
03

Test if arctangent is an odd function

Next, we'll test if arctangent is an odd function by checking if arctan(-x) = -arctan(x). From our work in Step 2, we have: tan(-arctan(x)) = -x Now, let's take the inverse tangent of both sides: -arctan(x) = arctan(-x) This condition holds true, which means arctan(x) is indeed an odd function.
04

Conclusion

After testing both the even and odd conditions, we find that arctangent function (tan^(-1)(x) or arctan(x)) is an odd function, since it satisfies the condition f(-x) = -f(x).

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