Function Problem

Hi Rich,

Thanks for the reply. So you are saying that the problem is asking to find which equation in the answers is equal to 1-x ?

If so, then I do not understand why D f(x) = x²(1 - x)² is the correct answer.

If you plug in 1 for the 1st equation it is 1-1 = 0
In 2nd equation it is 1²(1-1)² = 1(0) = 0

If you plug in 0 for the 1st equation it is 1-0 = 1
In the 2nd equation it is 0²(1-0)² = 0(1) = 0

So how does this work? Is my math or understanding incorrect?

Sincerely,

Melanie

Thanks Rich, Does my work below look correct?

1st equation (plug in 1):

f(x) = 1-x

1 - 1 = 0

2nd equation (plug in 1);

f(x) = x²(1-x)²

(1-x)²[1 - (1-x)]²

(1-1)²[1 - (1-1)]²

(0)²(1 - 0)²

0 * 1

= 0

me_1234 wrote:Can someone please explain this problem to me? Thank you!

For which of the following functions f is f(x) = f(1-x) for all x?

f(x) = 1 - x

f(x) = 1 - x²

f(x) = x² - (1 - x)²

f(x) = x²(1 - x)²

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