Question 191149
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A function that is decreasing on x > 0, y-intercept of 1, no x-intercept, and asymptotic to the x-axis (i.e. y = 0),


domain *[tex \LARGE \{x\ |\ x\ \in\ \R\ \}], range *[tex \LARGE \{y\ |\ y\ >\ 0,\ y\ \in\ \R\ \}]  is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x) = a^{-x}]


where *[tex \LARGE a] is any real number > 0, in other words a reflection of the general exponential function about the y-axis.


But you are looking for *[tex \LARGE a] such that *[tex \LARGE a^{-1} = 0.5 = {1 \over 2}] and that number would be 2.



*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x) = 2^{-x}]


The only thing that bothers me about this is the statement that f(x) is decreasing on the interval x > 0.  In fact, the function is decreasing across its entire domain.  That doesn't make this wrong, because the parameters of the problem don't specify what the function has to be doing on x <= 0.  But it makes me wonder what the question writer was thinking.  For that matter, why did the writer specify no x-intercepts when the range is restricted to values greater than zero?


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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