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 :

To find the -intercepts, substitute in the function.

.

Therefore the -intercept is .

To find the -intercepts, substitute in the function.

Therefore the -intercept is .

(C)

Symmetry :

Substitute in the function.

.

Hence .

Therefore the function is an odd function.

(D)

Asymptotes :

Horizontal asymptote :

Therefore the horizontal asymptote is .

Vertical asymptote :

Vertical asymptote appears when the function is not defined.

Therefore the vertical asymptote are and .

(E)

Intervals of increase or decrease : 

The function is .

Differentiate on each side with respect to .

is never zero and the function is negative for the domain of .

Therefore is decreasing over its domain.

(F)

Local Maximum and Minimum values :

From (E) it is clear that the function is only decreasing.

Therefore is no local minimum or maximum values.

(G)

Concavity and point of inflection :

.

Differentiate on each side with respect to .

Find inflection point by equating to zero.

Inflection point is.

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

Split the intervals into , , and .

\ Interval \	Test Value	Sign of	Concavity
		\ \	Down
		\ \	\ \ Up \
		\ \   \	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) -intercept is and -intercepts are .

(C) No symmetry.

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

(E) Decreasing on  , and .

(F) There is neither local minimum nor local maximum.

(G) Concave up on and .

      concave down on and .

(H) Graph of the function  is