If function f(x) satisfies f(1 - x) = f(2 + x) for all x, which of the

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Competition Mode Question

If function f(x) satisfies  f(1 - x) = f(2 + x)  for all x, which of the following can be true?

A.  f(x) = 3-x

B.  f(x) = 1-x^2

C.  f(x) = 2 - x - x^2

D.  f(x) = 2x - (1-x)^2

E.  f(x) = \frac{1-x}{2 + x}

madgmat2019 · Answer

A. f(x)=3−xf(x)=3−x ....

B. f(x)=1−x2f(x)=1−x2

C. f(x)=2−x−x2f(x)=2−x−x2

D. f(x)=2x−(1−x)2f(x)=2x−(1−x)2.....after substitution.....OA

E. f(x)=1−x2+x

freedom128 · Answer

f(1-x)=f(2+x) means that
x=0 --> f(1)=f(2)
x=1 --> f(-1)=f(), 
x=2 --> f(-1)=f(4), etc
It is clear that f(x) has to be symmetrical at x=1.5.

A. f(x)=3−x
NOT symmetrical at x=1.5

B. f(x)=1−x^2
NOT symmetrical at x=1.5. In fact, f(x) is symmetrical at x=0.

C. f(x)=2−x−x^2=(2+x)(1-x)
NOT symmetrical at x=1.5. In fact, f(x) is symmetrical at x=-0.5.

D. f(x)=2x−(1−x)^2= 2x-(x^2-2x+1)=-(x^2+1)
NOT symmetrical at x=1.5. In fact, f(x) is symmetrical at x=0.

E. f(x)=(1−x)^2/(2+x)
Symmetrical at x=1.5

Correct answer is (E)

unraveled · Answer

If function f(x) satisfies  f(1 − x) = f(2 + x)  for all x, which of the following can be true?

Looking at the options we can eliminate B, C and D.

Checking for option A gives  f(1 - x) = 2 + x  and  f(2 + x) = 1 – x .

Option E gives  f(1 − x) ≠ f(2 + x)

A.  f(x) = 3 − x

B.  f(x) = 1 − x^2

C.  f(x) = 2 – x − x^2

D.  f(x) = 2x − (1 − x)^2

E.  f(x) = \frac{(1 − x)}{(2 + x)}

Answer (A).

nishthagupta · Answer

this problem can be solved by simple number picking: plug in numbers.

Let's take x=2

1) f(x)=3−x 
-> 3-2=1

Now apply x=1 in the question stem 
f(1−x)=f(2+x)
-> 1-1 =0 and 2=1+3 These two are obviously not equal so A is not the right option

B) f(x)=1−(x)^2
By plugin,
1- (2)^2 = -3
Now putting this in the question stem
f(1−x)=f(2+x)
1-(-3)=4 and 2+ (-3) = -1

Continuing this for every option we arrive on the right answer -

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