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)=cos^(-1)x\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\n\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 (odd)\nb. symmetric with respect to the origin (even)\nc. 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.)\nd. no symmetry

analyze the function for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior f(x)=cos^(-1)x\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\n\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 (odd)\nb. symmetric with respect to the origin (even)\nc. 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.)\nd. no symmetry

Answer

Explanation:

Step1: Recall properties of inverse - cosine function

The inverse - cosine function (y = \cos^{-1}x) has domain ([- 1,1]) and range ([0,\pi]). Its derivative is (y'=-\frac{1}{\sqrt{1 - x^{2}}}), for (x\in(-1,1)). Since (y'<0) for (x\in(-1,1)), the function is always decreasing on its domain ([-1,1]).

Step2: Check for symmetry

Let (f(x)=\cos^{-1}x). Then (f(-x)=\cos^{-1}(-x)=\pi-\cos^{-1}x). Since (f(-x)\neq f(x)) and (f(-x)\neq - f(x)), the function is neither even nor odd. Also, there is no point of symmetry.

Answer:

For the increasing - decreasing behavior: B. The function is always decreasing For the symmetry: D. No symmetry