If `f:RtoR` and `g:RtoR` are difined by `f(x)=|x|` and `g(x)=[x-3]` for `x inR`, then `g(f(x)):-(8)/(5)ltxlt(8)/(5)` is equal to

Options

A. `{0,1}`
B. `{1,2}`
C. `{-3,-2}`
D. `{2,3}`

Answer

To solve the problem, we need to find the value of \( g(f(x)) \) for \( -\frac{8}{5} < x < \frac{8}{5} \), where \( f(x) = |x| \) and \( g(x) = [x - 3] \) (the greatest integer function).

### Step-by-step Solution:

1. **Identify the Functions**:
- We have \( f(x) = |x| \).
- We have \( g(x) = [x - 3] \).

2. **Find the Range of \( f(x) \)**:
- Since \( x \) is in the interval \( -\frac{8}{5} < x < \frac{8}{5} \), the output of \( f(x) \) will be:
\[ f(x) = |x| \implies 0 \leq f(x) < \frac{8}{5} \]

3. **Substituting \( f(x) \) into \( g(x) \)**:
- We need to find \( g(f(x)) = g(|x|) \).
- Thus, we need to evaluate \( g(|x|) = [|x| - 3] \).

4. **Determine the Range of \( |x| - 3 \)**:
- Since \( 0 \leq |x| < \frac{8}{5} \), we can find the range of \( |x| - 3 \):
\[ -3 < |x| - 3 < \frac{8}{5} - 3 \]
- Calculating \( \frac{8}{5} - 3 \):
\[ \frac{8}{5} - 3 = \frac{8}{5} - \frac{15}{5} = -\frac{7}{5} \]
- Therefore, the range of \( |x| - 3 \) is:
\[ -3 < |x| - 3 < -\frac{7}{5} \]

5. **Evaluate \( g(f(x)) = g(|x|) = [|x| - 3] \)**:
- The greatest integer function \( g(y) = [y] \) will take the value of \( -3 \) when \( -3 < y < -\frac{7}{5} \).
- Since \( -3 < |x| - 3 < -\frac{7}{5} \), we conclude:
\[ g(f(x)) = -3 \]

### Final Answer:
Thus, \( g(f(x)) = -3 \) for \( -\frac{8}{5} < x < \frac{8}{5} \).

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