Let g(x) = x - [x] - 1 and `f(x) = {{:(-1", " x lt 0),(0", "x =0),(1", " x gt 0):}` [.] represents the greatest integer function then for all x, f(g(x)) = .

Options

A. 2
B. 1
C. 0
D. `-1`

Answer

To solve the problem step by step, we will analyze the functions \( g(x) \) and \( f(x) \) and then find \( f(g(x)) \).

### Step 1: Define the functions
We have:
- \( g(x) = x - [x] - 1 \)
- \( f(x) = \begin{cases} -1 & \text{if } x < 0 \\ 0 & \text{if } x = 0 \\ 1 & \text{if } x > 0 \end{cases} \)

### Step 2: Understand the function \( g(x) \)
The function \( g(x) \) can be rewritten using the definition of the greatest integer function \( [x] \). The expression \( x - [x] \) gives the fractional part of \( x \), denoted as \( \{x\} \). Therefore, we can express \( g(x) \) as:
\[ g(x) = \{x\} - 1 \]
where \( \{x\} = x - [x] \).

### Step 3: Analyze the range of \( g(x) \)
The fractional part \( \{x\} \) always lies in the range \( [0, 1) \). Thus:
- The minimum value of \( g(x) \) occurs when \( \{x\} = 0 \), giving \( g(x) = 0 - 1 = -1 \).
- The maximum value occurs when \( \{x\} \) approaches 1, giving \( g(x) \) values approaching \( 0 - 1 = -1 \).

Thus, \( g(x) \) will always yield values in the range \( (-1, 0) \).

### Step 4: Evaluate \( f(g(x)) \)
Now we need to find \( f(g(x)) \). Since \( g(x) \) always results in values in the interval \( (-1, 0) \), we can apply the function \( f \):
- For \( g(x) < 0 \), according to the definition of \( f(x) \):
\[ f(g(x)) = -1 \]

### Conclusion
Thus, for all \( x \):
\[ f(g(x)) = -1 \]