This answer presumes that only one statement is untrue, as you did not include a line about "Circle all that apply."
In order to determine things such as whether the function is increasing or decreasing on a given interval, or its concavity, we must take the derivative of the function.
Since we have #e^(1/x)#, we must use the chain rule, which states that given #f(x) = g(h(x)), (df)/dx = (dh)/dx (dg)/(dh)#. With #g(h) = e^h, h(x) = 1/x, (dh)/(dx) = -1/x^2, (dg)/(dh) = e^h#, so...
#(df)/dx = -1/(x^2)e^(1/x)#
Looking at this function, from the interval #(-1/2, 0)# our x-value will be negative. With this being the case, some analysis shows us that the #e^(1/x)# component is greater than 0 for all x, by the definition of an exponential function. Further, #x^2>=0# for all x, so the denominator for #-1/x^2# is positive for all x, meaning that #-1/x^2<0# for all x. That means that #f'(x) <0# for all x, which in turn means the function is always decreasing. Thus, it is not increasing on #(-1/2, 0)#, and thus 3 is false.
