Question 824148
The key here is the word "must". This means that the answer must be true <i>no matter what a and b are</i> (as long as a < b).<br>
I'm not sure how anyone could think answers A or B must be true. They might be true for certain a's and b's but not for all possible a's and b's.<br>
Answer C is not correct either. A basic fact about exponents is that two powers of the same number are equal <i>only if the exponents are equal, too!</i>. Since a < b (not equal), (1/3)^a cannot be equal to (1/3)^b!<br>
Answer D looks good. After all, if b is greater than a then shouldn't (1/3)^b be greater than (1/3)^a? Answer: NO! 1/3 is a number between 0 and 1. And when you raise numbers like this to powers, <i>they get smaller!</i> Look at powers of 1/3:
(1/3)^2 = 1/9 (smaller than 1/3). (1/3)^3 = 1/27 (smaller than both 1/3 and 1/27). So D is not correct! (It would be correct if the function's base was a number greater than 1 instead of between 0 and 1.)<br>
Answer E is correct. (If you're not sure of this, look at the graph of f(x):
{{{graph(400, 400, -5, 5, -5, 5, (1/3)^x)}}}
Now pick <i>any</i> two points on this curve. The one farther to the left is (a, f(a)) since a < b. Which point is higher? No matter what two points you pick, the one on the left is higher. So f(a) > f(b) for all possible a's and b's.