Functions

pratyoosh wrote:Q. For which of the following functions 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)

Ans: D

Basically, f(x) = f(1 - x) means we need to find an option where if we put 1-x in the place of x the value of the equation will not change.

A. f(x) = 1 - x
f(1-x) = 1 - (1 - x) Wrong

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

Like this we can put and check and find out that D is correct. This is a lengthy but sure shot one but if we analyze the options mentally we can solve the same in no time.

f(x) = f(1 - x) should be satisfied. Thus, when u replace x by 1-x we will get the same equation.

See only option D takes the form of f(x) = x^2 * (1-x)^2. Here x and 1-x both have same degree of exponent and both are multiplied. Thus, interchangeable.

pratyoosh wrote:Q. For which of the following functions 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)

Ans: D

To solve this start by plugging x=1-x in the option answers.

1) f(x) = 1-x, f(1-x) should also give the same value i.e. 1-x when x=1-x
f(1-x) = 1-(1-x) = 1-1+x = x. Not True, discard.

2) f(x) = 1-x^2, f(1-x) = 1- (1-x)^2 , this too doesn't work, discard.

3) f(1-x) = (1-x)^2 - [1-(1-x)^2] , this fails too.

4) f(x) = x^2(1-x)^2; f(1-x) = (1-x)^2 [1 - (1-x)]^2 = (1-x) ^2[1-1+x] ^2 = (1-x)^2(x)^2 = f(x). True. Pick D.