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.
Here denominator of the function .
Therefore, 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.
The function is defined for all real numbers.
Therefore, there are no vertical asymptotes.
(E)
Intervals of increase or decrease :
The function is .
Differentiate on each side with respect to .
.
Find critical points by equating .
and .
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 and .
(F)
Local Maximum and Minimum values :
The function has a local minimum at , because changes its sign from negative to positive.
Substitute in .
.
Local minmum is .
The function has a local maximum at , because changes its sign from positive to negative.
Substitute in .
.
Local maximum is .
(G)
Concavity and point of inflection :
.
Differentiate on each side with respect to .
.
Find inflection point by equating to zero.
and
, and .
Inflection points are , and .
Test intervals into , , and .
Interval |
Test Value | Sign of | Concavity |
|
Down | ||
|
Up |
||
|
Down | ||
Up |
The graph is concave up in the interval and .
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) The function is an odd function.
(D) The horizontal asymptote is and there is no vertical asymptotes.
(E) The graph is increasing in the interval .
The graph is decreasing in the interval .
(F)
Local minmum is .
Local maximum is .
(G)
The graph is concave up in the interval and .
The graph is concave down in the interval and .
(H) Graph of the function is
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