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. the function has one horizontal asymptote at.\n(type an equation. simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the equation.)\nb. the function has two horizontal asymptotes. the top asymptote is y = π/2 and the bottom asymptote is y=-π/2.\n(type equations. simplify your answers. type exact answers, using π as needed. use integers or fractions for any numbers in the equations.)\nc. the function has no horizontal asymptotes.\nidentify any vertical asymptotes. select the correct choice and, if necessary, fill in the answer box to complete your choice.\na. the function has two vertical asymptotes. the leftmost asymptote is and the rightmost asymptote is.\n(type equations. simplify your answers. type exact answers, using π as needed. use integers or fractions for any numbers in the equations.)\nb. the function has one vertical asymptote at.\n(type an equation. simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the equation.)\nc. the function has no vertical asymptotes.

analyze the function for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. f(x)=tan^(-1)x\na. the function has one horizontal asymptote at.\n(type an equation. simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the equation.)\nb. the function has two horizontal asymptotes. the top asymptote is y = π/2 and the bottom asymptote is y=-π/2.\n(type equations. simplify your answers. type exact answers, using π as needed. use integers or fractions for any numbers in the equations.)\nc. the function has no horizontal asymptotes.\nidentify any vertical asymptotes. select the correct choice and, if necessary, fill in the answer box to complete your choice.\na. the function has two vertical asymptotes. the leftmost asymptote is and the rightmost asymptote is.\n(type equations. simplify your answers. type exact answers, using π as needed. use integers or fractions for any numbers in the equations.)\nb. the function has one vertical asymptote at.\n(type an equation. simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the equation.)\nc. the function has no vertical asymptotes.

Answer

Answer:

  1. Horizontal asymptotes:
    • B. The function has two horizontal asymptotes. The top asymptote is $y = \frac{\pi}{2}$ and the bottom asymptote is $y=-\frac{\pi}{2}$.
  2. Vertical asymptotes:
    • C. The function has no vertical asymptotes.

Explanation:

Step1: Recall the properties of $y = \tan^{- 1}x$

The inverse - tangent function $y=\tan^{-1}x$ is the inverse of the tangent function restricted to the interval $(-\frac{\pi}{2},\frac{\pi}{2})$.

Step2: Analyze horizontal asymptotes

As $x\to+\infty$, $\tan^{-1}x\to\frac{\pi}{2}$, and as $x\to-\infty$, $\tan^{-1}x\to-\frac{\pi}{2}$. So, it has two horizontal asymptotes $y = \frac{\pi}{2}$ and $y =-\frac{\pi}{2}$.

Step3: Analyze vertical asymptotes

The domain of $y=\tan^{-1}x$ is $(-\infty,\infty)$. The function is continuous for all real - valued $x$. There are no values of $x$ for which the function approaches infinity or negative infinity, so there are no vertical asymptotes.