GATE IN 2023 | GA Question: 9

Question

Which one of the given figures $\text{P, Q, R}$ and $\text{S}$ represents the graph of the following function?

$$f(x)=|| x+2|-| x-1||$$

 

 

• A. $\text{P}$
• B. $\text{Q}$
• C. $\text{R}$
• D. $\text{S}$

Answer 1

lets first find the graph for $y = |x+2| \ – \ |x-1|$ and then we will just take absolute value of y i.e make all the negative y’s positive and get our answer
$$y = |x+2| \ – \ |x-1|$$ has critical points $-2,1$

for $$x\gt1 \rightarrow (x+2)-(x-1) = +3$$ $$(x\gt-2) \ \ \& (x \le 1) \rightarrow (x+2)-(-(x-1)) = 2x+1$$

$$x\le-2 \rightarrow -(x+2)-(x-1) = -3$$

now plotting the graph after getting the equations for $y = |x+2| \ – \ |x-1|$

now just take the mod and negative part will invert to positive

hence the answer is P

Answer 2

• A. Figure P depicts the correct graph. The critical points are -2 and 1.  So, we have three cases:
  1. x < -2 in which both (x+2) and (x-1) are negative. So after solving you will get 3.
  2. -2 <= x < 1 in which (x+2) is positive but (x-1) is negative. So, after solving you will get |2x+1|.
  3. x >= 1 in which both are positive and you will get 3 again.

Answer 3

Or also you can do option elimination by putting different value of X in f(x). 

so option is B can never be answer because f(x) is in mod so value of f(x) can never be negative which is shown in option B.

now Option A, C, D remaing for them just put value of X = -4 in f(x) and we get answer as 3 which shown only in Option A so that option A is answer other answer become eliminated.