Around the World in 80 Questions (Day 7): If f(x) = |x - 1| and

Question

If  f(x) = |x - 1|  and  g(x) = |x + 2| , then what is the range of all possible values of x such that  f(g(x)) = 0  ?

A. 0
B. 1
C. 2
D. 3
E. 4

Bunuel · Accepted Answer

Official Solution:

If  f(x) = |x - 1|  and  g(x) = |x + 2| , then what is the range of all possible values of  x  such that  f(g(x)) = 0 ?

A. 0
B. 1
C. 2
D. 3
E. 4

Given that  g(x) = |x + 2| , we have  f(g(x)) = f(|x + 2|) .

Given that  f(x) = |x - 1| , we have  f(|x + 2|) = ||x + 2| - 1| .

Thus, we are told that  ||x + 2| - 1|=0

This implies that  |x + 2| = 1 , which yields  x + 2 = 1  or  x + 2 = -1 . This simplifies to  x = -1  or  x = -3 .

Therefore, the range of all possible values of  x  is  -1 - (-3)=2

Answer: C

kratos0906 · Answer

My Answer: C

f(g(x)) = ||x+2|-1|
||x+2|-1|=0, 
|x+2|=1

So there are two option, x+2=1, x=-1
and x+2=-1, x=-3

So, range=-1-(-3)=2

NoobGmatTaker · Answer

f(g(x)) becomes f(|x+2|).

Now, we know that f(x) = |x-1|. So, f(|x+2|) = ||x+2|-1|.

We are given that f(g(x)) = 0. So, ||x+2|-1| = 0.

The absolute value of a number is always non-negative. It is zero only when the number itself is zero. So, |x+2|-1 = 0.

Solving this equation gives us x+2 = 1 or x+2 = -1.

So, x = -1 or x = -3.

Answer is C : 2

missionmba2025 · Answer

||x+2|-1| = 0

|p| is always non negative.

|x+2| - 1 = 0

|x+2| = 1

1) x + 2 = 1

x = -2 + 1 = -1

2) x + 2 = -1

x = -2 - 1 = -3

Two values are possible x = -1 and x = -3

IMO C

Oppenheimer1945 · Answer

fog(x)=||x+2|-1|;
graphically, fog(x)=0 gives 2 solutions x=-3,-1
Range =2 C

MohdZaidKhan · Answer

This is a bogus question and all the solutions too .. The f(g(x) = 0 => x= 2 values , there is no range of x which we get after substracting the roots .... range of x is 2 absolute values .  Bunuel .

Bunuel · Answer

Before jumping to conclusions, it’s important to study the solutions carefully. The “range” of solutions means the largest value minus the smallest value. Here, the maximum value is -1 and the minimum is -3, so the range is -1 - (-3) = 2. Please review the solution more carefully next time.

Around the World in 80 Questions (Day 7): If f(x) = |x - 1| and

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