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Function Problem

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by me_1234 » Tue Dec 09, 2014 1:42 pm
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?

Answer Choices:

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)
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by [email protected] » Tue Dec 09, 2014 1:53 pm
Hi melanie.espleland,

Functions are really just equations for graphs. Function questions can often be solved by TESTing VALUES or doing algebra.

Here, we're asked which of these functions has the same RESULT when you plug in (X) and (1-X). This is a perfect situation for TESTing VALUES.

Since we're dealing with functions/graphs, we can TEST any value for X that we want. Let's keep things super-easy and use X = 0.

So, we now need to plug in 0 and 1-0 = 1 into each equation and track the results...

Answer A:
X = 0 ===> 1
X = 1 ===> 0
Not the same result

Answer B:
X = 0 ===> 1
X = 1 ===> 0
Not the same result

Answer C:
X = 0 ===> 0 - 1 = -1
X = 1 ===> 1 - 0 = 1
Not the same result

Answer D:
X = 0 ===> 0(1) = 0
X = 1 ===> 1(0) = 0
SAME RESULT

Answer E:
X = 0 ===> 0/1
X = 1 ===> 1/0
Not the same result

Final Answer: D

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by me_1234 » Tue Dec 09, 2014 2:28 pm
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
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by [email protected] » Tue Dec 09, 2014 6:00 pm
Hi melanie.espeland,

Here's some information about how to interpret function "language":

If we're given:
f(X) = X + 10

Then you can rewrite the function as:
Y = X + 10
and then graph it.

With this same function, if we have:
f(3) = X + 10
then we have to plug 3 in for "x" to get an end result...
f(3) = 3 + 10 = 13

In this question, we're asked for:
f(X)
f(1-X)
which means that we have to plug in each of those values wherever we see an "x"; the question asks us to find the function that has the SAME end result when you plug in EACH of those values. Given the relative complexity of some of the answer choices, it's easier to TEST Values than to plug in (X) and (1-X) and do all that algebra.

With the example I used (X = 0), so we're looking for:
f(0)
f(1)

Only Answer D gives us the same end result for both.

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by me_1234 » Tue Dec 09, 2014 6:26 pm
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
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by Jeff@TargetTestPrep » Wed Dec 13, 2017 11:03 am
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

Jeffrey Miller
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