study wrote:
For which of the following functions f is f(x) = f(1-x) for all 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)
Since we are not given any restrictions on the value of x, let’s let x = 1. Thus, we are determining for which of the following functions is f(1) = f(1-1), i.e., f(1) = f(0). Next, we can test each answer choice using our value x = 1.
A. f(x) = 1 - x
f(1) = 1 - 1 = 0
f(0) = 1 - 0 = 1
Since 0 does not equal 1, A is not correct.
B. f(x) = 1 - x^2
f(1) = 1 - 1^2 = 1 - 1 = 0
f(0) = 1 - 0^2 = 1 - 0 = 1
Since 0 does not equal 1, B is not correct.
C. f(x) = x^2 - (1 - x)^2
f(1) = 1^2 - (1 - 1)^2 = 1 - 0 = 1
f(0) = 0^2 - (1 - 0)^2 = 0 - 1 = -1
Since 1 does not equal -1, C is not correct.
D. f(x) = x^2*(1 - x)^2
f(1) = 1^2*(1 - 1)^2 = 1(0)= 0
f(0) = 0^2*(1 - 0)^2 = 0(2) = 0
Since 0 equals 0, D is correct.
Alternate Solution:
Let’s test each answer choice using x and 1 - x.
A. f(x) = 1 - x
f(x) = 1 - x
f(1 - x) = 1 - (1 - x) = x
Since 1 - x does not equal x, A is not correct.
B. f(x) = 1 - x^2
f(x) = 1 - x^2
f(1 - x) = 1 - (1 - x)^2 = 1 - (1 + x^2 -2x) = 2x - x^2
Since 1 - x^2 does not equal 2x - x^2, B is not correct.
C. f(x) = x^2 - (1 - x)^2
f(x) = x^2 - (1 - x)^2 = x^2 - (1 + x^2 - 2x) = 2x - 1
f(1 - x) = (1 - x)^2 - (1 - (1 - x))^2 = 1 + x^2 - 2x - x^2 = 1 - 2x
Since 2x - 1 does not equal 1 - 2x, C is not correct.
D. f(x) = x^2*(1 - x)^2
f(x) = x^2*(1 - x)^2
f(1 - x) = (1 - x)^2*(1 - (1 - x))^2 = (1 - x)^2*x^2
Since x^2*(1 - x)^2 equals (1 - x)^2*x^2, D is correct.
Answer: D
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