The function is $\"\"$ .

(A)

Domain :

The function is $\"\"$ .

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

Function inside the logarithm should be greater than zero.

$\"\"$

$\"\"$ and $\"\"$

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.

$\"\"$

$\"\"$ and $\"\"$

$\"\"$ and $\"\"$ .

(C)

Symmetry :

Substitute $\"\"$ in the function.

$\"\"$

Here $\"\"$

Therefore the function $\"\"$ is neither odd nor even.

(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.

Thus, the critical points exist when $\"\"$ .

Equate $\"\"$ to zero.

$\"\"$

$\"\"$ is not in the domain of the function, hence it is not considered.

No critical points.

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

\ \

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

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

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

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

(F)

Local Maximum and Minimum values :

The function $\"\"$ has no critical points, hence the function has no extrema.

(G)

Concavity and point of inflection :

$\"\"$ .

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

$\"\"$

$\"\"$ .

Find the inflection points.

Equate $\"\"$ to zero.

$\"\"$

$\"\"$ and $\"\"$ .

Imaginary values of $\"\"$ are not considered, hence no inflcetion points.

Concavity:

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

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

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

The function is concave down in the interval $\"\"$ .

$\"\"$

(H)

Graph :

Graph of the function $\"\"$ :

$\"\"$

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

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

(D) No asymptotes.

(E)

Increasing on $\"\"$ .

Decreasing on $\"\"$ .

(F)

No extrema points.

(G)

No Inflection point.

The function is concave down in the interval $\"\"$ .

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

$\"\"$