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Chapter 4: Problem 60

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

Short Answer

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

Step by step solution

01

Understanding the Graph

Graph the function \( g(x) = \csc(x) \) using a graphing tool. The graph will reveal that the function is neither symmetrical about the y-axis nor origin. Thus, it does not fall into the category of an even or odd function based on the visual analysis.
02

Verification using mathematical definitions

Verify algebraically whether the function is even or odd using the definitions. For an even function, evaluate \( g(-x) \) and see if it equals \( g(x) \). For an odd function, evaluate \( g(-x) \) and see if it equals -\( g(x) \). For \( g(x)= \csc(x) \), \( g(-x) = \csc(-x) = -\csc(x) \) due to the property of sine that \(\sin(-x) =-\sin(x)\), but it does not equal \( \csc(x) \), contrary to the condition for evenness. Also, \( g(-x)= -\csc(x) \) equals -\( g(x) \) only for certain x in the domain, but not all, which contradicts the condition for oddness.
03

Final Conclusion

As per the mathematical definitions and graph examination, the function \( g(x) = \csc(x) \) is neither even nor odd.

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