If f(x) = (x - 1)^2 + 3, which of the following is true for all x?

Question

If  f(x) = (x - 1)^2 + 3 , which of the following is true for all x?

I.  2f(x) = f(x - 1) + f(x + 1)

II.  f(2 - x) = f(x)

III.  f(x) = f(-x)

A. I only

B. II only

C. III only

D. None

E. I, II and III

greenisthecolor · Answer

B.

I started with III, since it seemed the easiest to find and that wasn't true. 
 f(-x) = (-x-1)^2 + 3 = x^2+1+2x+3 = x^2 + 2x + 4  NOT equal to  f(x)

So E is out.

Then took II.
 f(2-x) = (2-x-1)^2 + 3 = (-x+1)^2 + 3  will be equal to  f(x) . Thus, B.

Is there any shorter approach?

PKN · Answer

Given,  f(x) = (x - 1)^2 + 3

I.  f(x - 1) + f(x + 1)=(x-2)^2+3+x^2+3 = 2(x^2-2x+5)=2((x-1)^2+3))+2 = 2f(x)+2
eq{2f(x)} 
II.  f(2 - x)=(2-x-1)^2+3=(1-x)^2+3=(x-1)^2+3 = f(x) 
III.  f(-x)=(-x-1)^2+3=x^2+2x+1+3=x^2+2x+4
eq{f(x)}

Ans. (B)

manjot123 · Answer

gmatbusters  can we do this question by putting values of x?

GMATBusters · Answer

Hi Substituting values of x to prove that an equation is valid is tricky because sometimes an equation valid for one value of x doesn't satisfy other value of x. Always use substitute value of x to prove that equation is not valid, it works all the time.

For example: in the given question, III. f(x) = f(-x) , it is valid for x = 0, but not for x = 2. hence substituting values is not a good approach for this question.

BrushMyQuant · Answer

↧↧↧ Detailed Video Solution to the Problem ↧↧↧

https://www.youtube.com/watch?v=6QxcN807yII

Let's solve the problem using two methods

Method 1: Algebra

Given that  f(x) = (x - 1)^2 + 3

I.  2f(x) = f(x - 1) + f(x + 1)

To find f(x-1) we will replace x with x - 1 in  f(x) = (x - 1)^2 + 3  
=>  f(x-1) = (x - 1 - 1)^2 + 3 = (x - 2)^2 + 3 = x^2 - 2*2x + 2^2 + 3 = x^2 - 4x + 7

Similarly,  f(x+1) = (x + 1 - 1)^2 + 3 = x^2 + 3  
=>  f(x-1) + f(x+1)  =  x^2 - 4x + 7 + x^2 + 3  =  2x^2 - 4x + 10

2f(x) =  2 * ((x - 1)^2 + 3)  =  2 * ((x^2 - 2x + 1)+ 3)  =  2x^2 - 4x + 8 
Clearly,  2x^2 - 4x + 8   ≠   2x^2 - 4x + 10  
=>   FALSE

II.  f(2 - x) = f(x)   
 f(2-x) = (2 - x - 1)^2 + 3 = (1 - x)^2 + 3 = (x - 1)^2 + 3 = f(x)  
=>   TRUE

III.  f(x) = f(-x)   
 f(-x) = (-x - 1)^2 + 3 = (x + 1)^2 + 3 = x^2 + 2x + 1 + 3 = x^2 + 2x + 4  
Clearly,   x^2 + 2x + 4    ≠    x^2 - 2x + 4   
=>   FALSE

So,  Answer will be B

Method 2 Substitution

Lets put x = 1 and check which all options are not true

I.  2f(x) = f(x - 1) + f(x + 1)  
2f(1) = f(1-1) + f(1+1) = f(0) + f(2)

f(x) = (x - 1)^2 + 3  
=>  f(0) = (0 - 1)^2 + 3 = 1 + 3 = 4 
=>  f(1) = (1 - 1)^2 + 3 = 0 + 3 = 3  
=>  f(2) = (2 - 1)^2 + 3 = 1 + 3 = 4

=> f(0) + f(2) = 4 + 4 = 8
=> 2f(1) = 2*3 = 6
Clearly, 6  ≠  8
=>   FALSE   
 
 II.  f(2 - x) = f(x)   
f(2 - 1) = f(1)
=> f(1) = f(1)
Now, this is   TRUE   for x = 1, but we will have to check for other values also to be very sure

Let's put x = 0 
f(2 - 0) = f(0)
=> f (2) = f(0)
We calculated above that both are equal to 4
=>   TRUE

Let's put x = 99 
f(2 - 99) = f(99)
=> f (-97) = f(99)
=>  f(-97) = (-97 - 1)^2 + 3 = 98^2 + 3 
=>  f(99) = (99 - 1)^2 + 3 = 98^2 + 3  
=>   TRUE

III.  f(x) = f(-x)   
=> f(1) = f(-1)
=>  f(-1) = (-1 - 1)^2 + 3 = 4 + 3 = 7 
And f(1) = 3
Clearly, 3  ≠  7
=>   FALSE
  
So,  Answer will be B 
Hope it helps!

Watch the following video to MASTER Functions and Custom Characters

https://www.youtube.com/watch?v=b7JoMX0gk8M

If f(x) = (x - 1)^2 + 3, which of the following is true for all x?

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If f(x) = (x - 1)^2 + 3, which of the following is true for all x?

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