The function is .

(A)

Domain :

The function is .

The function is a logarithm function hence it is continuous for .

Therefore the domain of the function is

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 -intercepts are .

(C)

Symmetry :

Substitute in the function.

Here

Therefore the function is symmetric about the origin..

(D)

Asymptotes :

Horizontal asymptote :

There is no horizontal asymptote.

Vertical asymptote :

There is vertical asymptote.

(E)

Intervals of increase or decrease :

The function is .

Differentiate on each side with respect to .

Find the critical points.

The critical points exist when .

Equate  to zero.

The critical points are and.

The test intervals are and.

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

The function is increasing on the intervals .

The function is decreasing on the interval .

(F)

Local Maximum and Minimum values :

The function has a local maximum at ,

Substitute in .

Local maximum is .

(G)

Concavity and point of inflection :

Differentiate on each side with respect to .

Find the inflection points.

Equate to zero.

The inflection point is

Substitute in .

The test intervals are and .

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

The graph is concave up in the interval .

The graph is concave down in the interval .

The inflection point is .

Graph :

Graph of the function  :