Let `f (x) =-1 +|x-2| and g (x) =1-|x|` then set of all possible value (s) of for which (fog) (x) is discontinuous is:

Options

A. `{0,1,2}`
B. `{0,2}`
C. `{0}`
D. an empty set

Answer

To solve the problem, we need to find the set of all possible values for which the composition \( (f \circ g)(x) \) is discontinuous. We start with the given functions:

1. \( f(x) = -1 + |x - 2| \)
2. \( g(x) = 1 - |x| \)

### Step 1: Find \( g(x) \)

The function \( g(x) = 1 - |x| \) is defined for all real numbers \( x \). It is a piecewise function:

- For \( x \geq 0 \): \( g(x) = 1 - x \)
- For \( x < 0 \): \( g(x) = 1 + x \)

### Step 2: Find \( f(g(x)) \)

Now we need to substitute \( g(x) \) into \( f(x) \):

\[ f(g(x)) = f(1 - |x|) = -1 + |(1 - |x|) - 2| \]

This simplifies to:

\[ f(g(x)) = -1 + |1 - |x| - 2| = -1 + | -1 - |x| | = -1 + | -1 - |x| | \]

### Step 3: Simplify \( f(g(x)) \)

Now we need to analyze \( | -1 - |x| | \):

- For \( |x| \geq 1 \): \( -1 - |x| \) is negative, so \( | -1 - |x| | = -(-1 - |x|) = 1 + |x| \)
- For \( |x| < 1 \): \( -1 - |x| \) is negative, so \( | -1 - |x| | = -(-1 - |x|) = 1 + |x| \)

Thus, we can write:

\[ f(g(x)) = -1 + (1 + |x|) = |x| \]

### Step 4: Analyze the continuity of \( f(g(x)) \)

The function \( f(g(x)) = |x| \) is continuous everywhere on the real line.

### Conclusion

Since \( f(g(x)) = |x| \) is continuous for all \( x \), there are no points of discontinuity.

### Final Answer

The set of all possible values for which \( (f \circ g)(x) \) is discontinuous is the empty set:

\[ \text{Set of discontinuous points} = \emptyset \]