QQuestionMathematics
QuestionMathematics
Which statement is true about y=arctan(x)?
A. The domain is (−π/ 2, π/ 2). B. The range is all real numbers.
C. The graph contains vertical asymptotes.
D. The graph is symmetric about the origin.
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Answer
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Step 1:Let's solve this step by step:
Step 2:: Understand the arctan function
The arctan (or inverse tangent) function, $$y = \arctan(x)$$, is a fundamental trigonometric function with specific characteristics.
Step 3:: Analyze the domain
- This eliminates option A, which incorrectly states the domain as $$(-\frac{\pi}{2}, \frac{\pi}{2})
This means you can input any real number into the function.
Step 4:: Analyze the range
The range of $$\arctan(x)$$ is $$(-\frac{\pi}{2}, \frac{\pi}{2})
- This means the output values are always between -\frac{\pi}{2} and \frac{\pi}{2}) - This eliminates option B, which claims the range is all real numbers
Step 5:: Check for vertical asymptotes
\arctan(x)$$ does NOT have vertical asymptotes
- The function is continuous for all real inputs - This eliminates option C
Step 6:: Symmetry analysis
The graph of $$\arctan(x)$$ is symmetric about the origin
- If f(x) = \arctan(x), then f(-x) = -f(x) - This confirms option D is true
Final Answer
The graph is symmetric about the origin. Key insights: - \arctan(x) is a continuous, odd function - Its range is (-\frac{\pi}{2}, \frac{\pi}{2}) - It passes through the origin with a slope of 1 - Symmetric about the origin means \arctan(-x) = -\arctan(x)
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