# functions-and-graphs-3

(3)

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.

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

(d)

f(x) = cosec x

Domain R - {n | n I}

Range R - (- 1, 1)

n = n n I is asymptote to y = cosec x

(e)

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)

f(x) = cot x

Domain R - {n | n I}

Range R

It has x = n n I as asymptotes.

(i)

where

x [-1, 1]

and y

As the graph of f

(ii)

Here,

Domain [-1, 1]

Range [0, ]

(iii)

Here ,

Domain R

Range .

(iv)

We know that the function f:(0, ) R, given by f() = cot is invertible.

Thus, domain of cot

(v)

The function f : [0, ] - (- , - 1) [1, ] given by f() = sec is invertible.

y = sec

(vi)

As we know, f: - {0} (- 1, 1) is invertible given by f() = cos.

y = cosec

Range - {0}>

Domain x [-1, 1]

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

Sketch y = sin(sin

when x [-1, 1] & y = x

Domain x [-1, 1]

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

Domain, x R

Range y = x y r

We should sketch

y = tan(tan~~v~~ x R

Domain x R - (-1, 1)

Range y = x y R - (-1, 1)

y = cosec(cosec

when x (-, -1) (1, )

__:__**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.

__: What is asymptotes ?__**Dumb Question**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**^{-1}x;Here,

Domain [-1, 1]

Range [0, ]

(iii)

**Graph of y = tan**^{-1}x;Here ,

Domain R

Range .

(iv)

**Graph of y = cot**^{-1}x;We know that the function f:(0, ) R, given by f() = cot is invertible.

Thus, domain of cot

^{-1}x R and Range (0, ).(v)

**Graph for y = sec**^{-1}x;The function f : [0, ] - (- , - 1) [1, ] given by f() = sec is invertible.

y = sec

^{-1}x, has domain R - (- 1, 1) and range [0, ] - : shown as(vi)

**Graph for y = cosec**^{-1}x;As we know, f: - {0} (- 1, 1) is invertible given by f() = cos.

y = cosec

^{-1}x; domain R - (- 1, 1)Range - {0}>

**Sketch of y = sin(sin**:^{-1}x)Domain x [-1, 1]

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

Sketch y = sin(sin

^{-1}x) onlywhen x [-1, 1] & y = x

**Sketch of curve y = cos(cos**:^{-1}x)Domain x [-1, 1]

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

**Sketch of curve y = tan(tan**:^{-1}x)Domain, x R

Range y = x y r

We should sketch

y = tan(tan

^{-1}x) = x**Sketch of curve y = cosec(cosec**:^{-1}x)Domain x R - (-1, 1)

Range y = x y R - (-1, 1)

y = cosec(cosec

^{-1}x) = x onlywhen x (-, -1) (1, )