analyze the function for domain, range, continuity, increasing or decreasing behavior, symmetry…

analyze the function for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. f(x)=tan^(-1)x\na. symmetric with respect to the origin (even)\nb. neither odd nor even, but symmetric with respect to the point (type an ordered pair. simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nc. symmetric with respect to the origin (odd)\nd. no symmetry\nthe function is bounded.\nidentify any local maxima. select the correct choice and, if necessary, fill in the answer box to complete your choice.\na. there is a maximum at x= (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nb. there are no local maxima.

analyze the function for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. f(x)=tan^(-1)x\na. symmetric with respect to the origin (even)\nb. neither odd nor even, but symmetric with respect to the point (type an ordered pair. simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nc. symmetric with respect to the origin (odd)\nd. no symmetry\nthe function is bounded.\nidentify any local maxima. select the correct choice and, if necessary, fill in the answer box to complete your choice.\na. there is a maximum at x= (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nb. there are no local maxima.

Answer

Answer:

  1. C. Symmetric with respect to the origin (odd)
  2. B. There are no local maxima

Explanation:

Step1: Recall the property of odd - functions

For a function (y = f(x)), if (f(-x)=-f(x)), it is odd. For (y = \tan^{- 1}x), we have (\tan^{-1}(-x)=-\tan^{-1}(x)). So it is symmetric about the origin.

Step2: Analyze the derivative of (y = \tan^{-1}x)

The derivative of (y=\tan^{-1}x) is (y'=\frac{1}{1 + x^{2}}). Since (y'=\frac{1}{1 + x^{2}}>0) for all (x\in(-\infty,\infty)), the function is always increasing. A function that is always increasing has no local maxima.