I came across this problem while solving the GMAT Prep
Question: 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)
Answer: D
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!!
For which of the following functions f(x) = f (1-x)?
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let x = 1
{A}
f(x) = 0
f(1-x) = 1 - (1-1) = 1
NO
{B}
f(x) = 1 - (1)^2 = 0
f(1-x) = 1 - (1-x)^2 = 1
NO
{C}
f(x) = 1 - 0 = 1
f(1-x) = (1-1)^2 - (1 - 1 + 1)^2 = 0 - 1 = -1
NO
{D}
f(x) = 1(1-1)^2 = 0
f(1-x) = (1-1)^2 (1-1+1)^2 = 0
YES
{E}
f(x) = 1/0 =ND
f(1-x) = 0
NO
Answer [spoiler]{D}[/spoiler]
{A}
f(x) = 0
f(1-x) = 1 - (1-1) = 1
NO
{B}
f(x) = 1 - (1)^2 = 0
f(1-x) = 1 - (1-x)^2 = 1
NO
{C}
f(x) = 1 - 0 = 1
f(1-x) = (1-1)^2 - (1 - 1 + 1)^2 = 0 - 1 = -1
NO
{D}
f(x) = 1(1-1)^2 = 0
f(1-x) = (1-1)^2 (1-1+1)^2 = 0
YES
{E}
f(x) = 1/0 =ND
f(1-x) = 0
NO
Answer [spoiler]{D}[/spoiler]
R A H U L
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When posting questions, please use the spoiler function to hide the correct answer. This will allow others to attempt the question without seeing the final answer.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!!
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
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Note: f(1-x) = -f(x-1) (unless squared)
If f(x) is our starting graph, then -f(x-1) is a vertical reflection that is translated to the right one unit.
What shape can be reflected vertically and moved right without changing? Only f(x) = 0
As this is not on the list, it must be a square function:
Only (D) f(x) = x^2( 1-x)^2 is totally square.
Let's check it:
f(x) = x^2( 1-x)^2
Replace x with (1-x):
f(x-1) = (1-x)^2(1-(1-x))^2 = x^2( 1-x)^2
CORRECT!
If f(x) is our starting graph, then -f(x-1) is a vertical reflection that is translated to the right one unit.
What shape can be reflected vertically and moved right without changing? Only f(x) = 0
As this is not on the list, it must be a square function:
Only (D) f(x) = x^2( 1-x)^2 is totally square.
Let's check it:
f(x) = x^2( 1-x)^2
Replace x with (1-x):
f(x-1) = (1-x)^2(1-(1-x))^2 = x^2( 1-x)^2
CORRECT!