SOLUTION: If f(x) = (1/3)^x and a < b Which of the following must be true? A. f(a) + f(b) = 3 B. f(a) + 1/3 = f(b) C. f(a) = f(b) D. f(a)

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Question 824148: If f(x) = (1/3)^x and a < b
Which of the following must be true?
A. f(a) + f(b) = 3
B. f(a) + 1/3 = f(b)
C. f(a) = f(b)
D. f(a) < f(b)
E. f(a) > f(b)
Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
The key here is the word "must". This means that the answer must be true no matter what a and b are (as long as a < b).

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.

Answer C is not correct either. A basic fact about exponents is that two powers of the same number are equal only if the exponents are equal, too!. Since a < b (not equal), (1/3)^a cannot be equal to (1/3)^b!

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, they get smaller! 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.)

Answer E is correct. (If you're not sure of this, look at the graph of f(x):

Now pick any 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.

SOLUTION: If f(x) = (1/3)^x and a < b Which of the following must be true? A. f(a) + f(b) = 3 B. f(a) + 1/3 = f(b) C. f(a) = f(b) D. f(a) < f(b) E. f(a) > f(b)