The function is .

(A)

Domain :

The function is .

All possible values of  is the domain of the function.

Denominator of the function should not be zero

The domain of the function  is .

(B)

Intercepts :

There is no -intercept, since  is not in  the domain.

To find the -intercepts, substitute  in the function.

Therefore the -intercept is .

(C)

Symmetry :

Substitute  in the function.

.

.

Therefore the function  is neither odd nor even.

(D)

Asymptotes :

Horizontal asymptote :

Therefore the horizontal asymptote is .

Vertical asymptote :

Vertical asymptote appears when the function is not defined.

Therefore the vertical asymptote is .

(E)

Intervals of increase or decrease : 

The function is .

Differentiate on each side with respect to .

.

Find critical points by equating .

.

Test intervals are based on  and where  is undefined.

Test intervals are ,  and .

\ Interval \	Test Value	Sign of	Conclusion
		\ \	Decreasing
		\ \	\   \ Increasing \
		\ \	Decreasing

 The graph is increasing in the interval .

The graph is decreasing in the interval , .

(F)

Local Maximum and Minimum values :

The function  has a local maximum at , because  changes its sign from positive to negative.

Substitute  in .

.

Local maximum is .

At , there is an vertical asymptote, so we donot consider local minimum at .

(G)

Concavity and point of inflection :

.

Differentiate  on each side with respect to .

.

Find inflection points by equating  to zero.

.

At  ,.

Inflection point is .

But at , the function is undefined,hence consider those points also.

Split the intervals into ,  and .

\ Interval \	Test Value	Sign of	Concavity
		\ \	Down
		\ \	\   \ Down \
		\ \	Up

The graph is concave up in the interval .

The graph is concave down in the interval  and .

(H)

Graph :

Graph of the function  :

(A) The domain of the function  is .

(B) No -intercept,  and -intercept is .

(C) No symmetry.

(D) The horizontal asymptote is  and the vertical asymptote is  .

(E) 

The graph is increasing in the interval .

The graph is decreasing in the interval , .

(F)

Local maximum is .

(G)

The graph is concave up in the interval .

The graph is concave down in the interval  and .

Inflection point is .

(H) Graph of the function   is

.