# Consider the function #f(x)=e^(-x^(2))#?

##
*Find the following*:

- domain;
*found:* #(-\infty,\infty)#
- x-intercepts;
*found:* #-x^2=-\infty# thus there is NO x-intercept
- y-intercept;
*found:* #y=1#
- symmetry
- vertical asymptotes;
*found:* NONE
- horizontal asymptotes;
*found:* #x=0# (exponential function rule?)
- intervals of increase and decrease
- local maxima and minima
- intervals of concavity
- inflection points

**If any characteristics are not present in the function, state "NONE". Then graph the function**

*I apologize for the confusion regarding the parts I can understand/answer! I have worked out which ones I actually need help with now.*

*Find the following*:

- domain;
*found:*#(-\infty,\infty)# - x-intercepts;
*found:*#-x^2=-\infty# thus there is NO x-intercept - y-intercept;
*found:*#y=1# - symmetry
- vertical asymptotes;
*found:*NONE - horizontal asymptotes;
*found:*#x=0# (exponential function rule?) - intervals of increase and decrease
- local maxima and minima
- intervals of concavity
- inflection points

**If any characteristics are not present in the function, state "NONE". Then graph the function**

*I apologize for the confusion regarding the parts I can understand/answer! I have worked out which ones I actually need help with now.*

*I apologize for the confusion regarding the parts I can understand/answer! I have worked out which ones I actually need help with now.*

##### 1 Answer

See below.

#### Explanation:

**Symmetry**

So the function is even and the graph of the function is symmetric with respect to the

**Increase/Decrease and Extrema**

Since

So

**Concavity and Inflection**

# = 2e^(-x^2)(2x^2-1)#

Analysis of the sign of

on

on

on

Therefore, there are two inflection points:

(Note that