(3) Tangent Function:

  f(x) = tan x
Domain R - {(2n + 1) }
Range R

y = tan x    increases strictly from - to as x increases from - to , to
x = ± , ± , .......... are asymptotes to y = tan x.


Dumb Question: What is asymptotes ?

Ans: A curve which is tangent to given curve at infinity.


(d) Coecant function:

  f(x) = cosec x
Domain R - {n | n I}
Range R - (- 1, 1)
n = n    n I is asymptote to y = cosec x



(e) Secant Function:

   f(x) = sec x
Domain R - {(2n + 1) | n I}
Range R - (- 1, 1)
x = (2n + 1), n I are asymptote to y = sec x.



(f) Cotangent Function:

  f(x) = cot x
Domain R - {n | n I}
Range R
It has x = n   n I as asymptotes.




Inverse Function:

(i) Graph of y = sin^-1 x;
    where
              x [-1, 1]
and         y


As the graph of f^-1 is mirror image of f(x) about y = x.

(ii) Graph of y = tan^-1x;
     Here,
              Domain [-1, 1]
              Range [0, ]



(iii) Graph of y = tan^-1x;

      Here ,
                Domain R
                Range .



(iv) Graph of y = cot^-1x;

     We know that the function f:(0, ) R, given by f() = cot is invertible.
     Thus, domain of cot^-1x R and Range (0, ).



(v) Graph for y = sec^-1x;

    The function f : [0, ] - (- , - 1) [1, ] given by f() = sec is invertible.
     y = sec^-1x, has domain R - (- 1, 1) and range [0, ] - : shown as



(vi) Graph for y = cosec^-1x;

     As we know,   f: - {0} (- 1, 1) is invertible given by f() = cos.
     y = cosec^-1x; domain R - (- 1, 1)
     Range - {0}>




Sketch of y = sin(sin^-1x):

Domain x [-1, 1]
and range y = x   y [-1, 1]
Sketch y = sin(sin^-1x) only
when x [-1, 1] & y = x




Sketch of curve y = cos(cos^-1x):

Domain x [-1, 1]
and range y = x   y [-1, 1]




Sketch of curve y = tan(tan^-1x):

Domain, x R
Range y = x   y r
We should sketch
y = tan(tan^-1x) = x v x R




Sketch of curve y = cosec(cosec^-1x):

Domain x R - (-1, 1)
Range y = x   y R - (-1, 1)
y = cosec(cosec^-1x) = x only
when x (-, -1) (1, )