The function is $\"\"$ .

(A)

Domain :

The function is $\"\"$ .

Function inside the logarithm value should be greater than zero.

$\"\"$ is always greater than zero.

All possible values of $\"\"$ is the domain of the function.

Therefore the domain of the function $\"\"$ is the set of all real numbers.

The domain of the function $\"\"$ is $\"\"$ .

(B)

Intercepts :

To find the $\"\"$ -intercepts, substitute $\"\"$ in the function.

$\"\"$

The $\"\"$ -intercept is $\"\"$ .

To find the $\"\"$ -intercepts, substitute $\"\"$ in the function.

$\"\"$

The $\"\"$ -intercept is $\"\"$ .

(C)

Symmetry :

Substitute $\"\"$ in the function.

$\"\"$

Here $\"\"$ , hence the function is a even function.

Therefore the function $\"\"$ is symmetric with respect to $\"\"$ axis.

(D)

Asymptotes :

There is no vertical asymptotes, since the function is continuous at all real numbers.

Horizontal asymptote :

No logarithmic function has horizontal asymptotes.

There is no horizontal asymptote.

(E)

Intervals of increase or decrease :

The function is $\"\"$ .

Differentiate $\"\"$ on each side with respect to $\"\"$ .

$\"\"$

Find the critical points.

Critical points exist when $\"\"$ .

Equate $\"\"$ to zero.

$\"\"$

The critical points are $\"\"$ .

The test intervals are $\"\"$ and $\"\"$ .

\ \

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

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

The function is decreasing on the intervals $\"\"$ .

The function is increasing on the interval $\"\"$ .

(F)

Local Maximum and Minimum values :

The function $\"\"$ has a local minimum at $\"\"$ , because $\"\"$ changes its sign from negative to positive.

Substitute $\"\"$ in $\"\"$ .

$\"\"$

Local minimum is $\"\"$ .

(G)

Concavity and point of inflection :

$\"\"$ .

Differentiate $\"\"$ on each side with respect to $\"\"$ .

$\"\"$

Find the inflection points.

Equate $\"\"$ to zero.

$\"\"$

Substitute $\"\"$ in $\"\"$ .

$\"\"$

Inflection point is $\"\"$ .

Substitute $\"\"$ in $\"\"$ .

$\"\"$

Inflection point is $\"\"$ .

The test intervals are $\"\"$ $\"\"$ and $\"\"$ .

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

\ Interval \	Test Value	Sign of $\"\"$	Concavity
$\"\"$	$\"\"$	\ $\"\"$ \	Concave Down
$\"\"$	$\"\"$	\ $\"\"$ \	\ \ Concave Up
$\"\"$	$\"\"$	\ $\"\"$ \	Concave Down

The graph is concave up in the interval $\"\"$ .

The graph is concave down in the interval $\"\"$ and $\"\"$ .

The inflection points are $\"\"$ and $\"\"$ .

$\"\"$

(H)

Graph :

Graph of the function $\"\"$ :

$\"\"$

(A) Domain of the function is $\"\"$ .

(B) $\"\"$ -intercept is $\"\"$ . and $\"\"$ -intercept is $\"\"$ .

(D) No asymptotes.

(E)

The function is decreasing on the intervals $\"\"$ .

The function is increasing on the interval $\"\"$ .

(F)

Local minimum is $\"\"$ .

(G)

The graph is concave up in the interval $\"\"$ .

The graph is concave down in the interval $\"\"$ and $\"\"$ .

The inflection points are $\"\"$ and $\"\"$ .

(H) Graph of the function is $\"\"$ .

$\"\"$ .