Let [x] denot the greatest integer `le x`. If `f (x)= [x] and g(x)=|x|` then the value of `f(g((8)/(5)))-g(f(-(8)/(5)))` is

Options

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

Answer

To solve the problem, we need to evaluate the expression \( f(g(\frac{8}{5})) - g(f(-\frac{8}{5})) \) where \( f(x) = [x] \) (the greatest integer less than or equal to \( x \)) and \( g(x) = |x| \) (the absolute value of \( x \)).

### Step-by-Step Solution:

1. **Calculate \( g(\frac{8}{5}) \)**:
\[ g\left(\frac{8}{5}\right) = \left|\frac{8}{5}\right| = \frac{8}{5} \]
Since \( \frac{8}{5} \) is positive, the absolute value remains the same.

2. **Calculate \( f(g(\frac{8}{5})) \)**:
\[ f(g(\frac{8}{5})) = f\left(\frac{8}{5}\right) = f(1.6) \]
To find \( f(1.6) \), we take the greatest integer less than or equal to \( 1.6 \):
\[ f(1.6) = [1.6] = 1 \]

3. **Calculate \( f(-\frac{8}{5}) \)**:
\[ f\left(-\frac{8}{5}\right) = f(-1.6) \]
To find \( f(-1.6) \), we take the greatest integer less than or equal to \( -1.6 \):
\[ f(-1.6) = [-1.6] = -2 \]

4. **Calculate \( g(f(-\frac{8}{5})) \)**:
\[ g(f(-\frac{8}{5})) = g(-2) = |-2| = 2 \]

5. **Combine the results**:
Now we substitute back into the original expression:
\[ f(g(\frac{8}{5})) - g(f(-\frac{8}{5})) = 1 - 2 = -1 \]

### Final Answer:
The value of \( f(g(\frac{8}{5})) - g(f(-\frac{8}{5})) \) is \( -1 \).