Problem Solving

sbhawal wrote: [/b] For which of the following functions f(x) = f (1-x)?
A. f(x) =1- x
B. f(x) = 1- x^2
C. f(x) = x^2 - (1-x)^2
D. f(x) = x^2( 1-x)^2
E. f(x) = x/(1-x)

Other than substituting values for x and checking each, what is the shorter alternative/method of arriving at the answer? It took me more than 5 mins to check all the options. I cannot use the same method during the exam. Pls help!!

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What numbers did you plug in? If you choose something like x = 7 or x = -5.4, it will take a while to evaluate each expression. However, if you choose a nice value for x (e.g., x = 0 or x = 1), then your calculations shouldn't take long at all.

Here's my solution:

Let's try plugging in an easy value for x. How about x = 0.
So, we can reword the question as, For which of the following functions is f(0)=f(1-0)
In other words, we're looking for a function such that f(0) = f(1)

A) f(x)=1-x
f(0)=1-0 = 1
f(1)=1-1 = 0
Since f(0) doesn't equal f(1), eliminate A

B) f(x) = 1 - x^2
f(0) = 1 - 0^2 = 1
f(1) = 1 - 1^2 = 0
Since f(0) doesn't equal f(1), eliminate B

C) f(x) = x^2 - (1-x)^2
f(0) = 0^2 - (1-0)^2 = -1
f(1) = 1^2 - (1-1)^2 = 1
Since f(0) doesn't equal f(1), eliminate C

D) f(x) = x^2(1-x)^2
f(0) = 0^2(1-0)^2 = 0
f(1) = 1^2(1-1)^2 = 0
Since f(0) equals f(1), keep D for now

E) f(x) = x/(1-x)
f(0) = 0/(1-0) = 0
f(1) = 1/(1-1) = undefined
Since f(0) doesn't equal f(1), eliminate E

Since only D satisfies the condition that f(x)=f(1-x) when x=0, the correct answer is D

Cheers,
Brent