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⁻¹x\n\na. the function is increasing on the interval and decreasing on the interval (type your answers in interval notation. use a comma to separate answers as needed.)\nb. the function is always decreasing.\nc. the function is always increasing.\ndescribe the symmetry of the function. select the correct choice and, if necessary, fill in the answer box to complete your choice.\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

analyze the function for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. f(x)=tan⁻¹x\n\na. the function is increasing on the interval and decreasing on the interval (type your answers in interval notation. use a comma to separate answers as needed.)\nb. the function is always decreasing.\nc. the function is always increasing.\ndescribe the symmetry of the function. select the correct choice and, if necessary, fill in the answer box to complete your choice.\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

Answer

Answer:

  1. C. The function is always increasing.
  2. C. Symmetric with respect to the origin (odd)

Explanation:

Step1: Analyze increasing - decreasing behavior

The derivative of $y = \tan^{- 1}x$ is $y'=\frac{1}{1 + x^{2}}$. Since $1+x^{2}>0$ for all real - valued $x$, then $y'>0$ for all $x\in(-\infty,\infty)$. So the function $y = \tan^{- 1}x$ is always increasing.

Step2: Analyze symmetry

Let $f(x)=\tan^{- 1}x$. Then $f(-x)=\tan^{- 1}(-x)=-\tan^{- 1}x=-f(x)$. By the definition of an odd function ($f(-x)=-f(x)$), the function $y = \tan^{- 1}x$ is symmetric with respect to the origin.