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There are two basic ways to approach this problem: 1. We can plug in numbers for x and 1-x, or 2. we can solve this one using algebra.
The former is probably slightly simpler in this case, so I'll go through those steps below:
Let's say x = 2, so we know that 1-x = -1
This means we want to find a situation where f(x) = f(1-x) or, using our new numbers, where f(2)=f(-1)
Here's the application of that in the answer choices:
A) f(x) = 1-x
f(2) = 1-2 = -1
f(-1) = 1 - (-1) = 2
These are not equal, so this isn't our answer.
B) f(x) = 1-x^2
f(2) = 1 - 2^2 = 1 - 4 = -3
f(-1) = 1 - (-1)^2 = 1 + 1 = 2
These are not equal, so this isn't our answer.
C) f(x) = x^2 - (1-x)^2
f(2) = 2^2 - (1-2)^2 = 4 - (-1)^2 = 4 - 1 = 3
f(-1) = (-1)^2 - (1-(-1))^2 = 1 - 2^2 = 1 - 4 = -3
These are not equal, so this isn't our answer.
D) f(x) = X^2(1-x)^2 (CORRECT)
f(2) = 2^2 * (1-2)^2 = 4 * (-1)^2 = 4 * 1 = 4
f(-1) = (-1)^2 * (1-(-1))^2 = 1 * 2^2 = 1*4 = 4
These two ARE equal, so this is our correct answer.
E) f(x) = x / (1-x)
f(2) = 2 / 1-2 = 2 / -1 = -2
f(-1) = -1 / (1-(-1)) = -1/2
These are not equal, so this isn't our answer.
Veritas Help
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