Question

Let \( [x] \) denote the greatest integer \( \leq x \). If \( f(x) = [x] \) and \( g(x) = |x| \), then the value of:
\[ f \left( g \left( \frac{8}{5} \right) \right) - g \left( f \left( \frac{-8}{5} \right) \right) \] is:

• A. \( 2 \)
• B. \( -2 \)
• C. \( 1 \)
• D. \( -1 \)

Hint:

The greatest integer function \( [x] \) returns the largest integer less than or equal to \( x \). The absolute function \( |x| \) removes the sign.

Answer 1

The Correct Option is D

Step 1: Calculate \( f(-8/5) \).
\[f \left( \frac{-8}{5} \right) = \left\lfloor \frac{-8}{5} \right\rfloor = -2.\]Step 2: Calculate \( g(8/5) \) and \( g(-8/5) \).
\[g \left( \frac{8}{5} \right) = \left| \frac{8}{5} \right| = \frac{8}{5}.\]\[g \left( \frac{-8}{5} \right) = \left| \frac{-8}{5} \right| = \frac{8}{5}.\]Step 3: Calculate \( f(g(8/5)) \) and \( g(f(-8/5)) \).
\[f \left( g \left( \frac{8}{5} \right) \right) = f \left( \frac{8}{5} \right) = \left\lfloor \frac{8}{5} \right\rfloor = 1.\]\[g \left( f \left( \frac{-8}{5} \right) \right) = g(-2) = | -2 | = 2.\]Step 4: Perform the final computation.
\[f \left( g \left( \frac{8}{5} \right) \right) - g \left( f \left( \frac{-8}{5} \right) \right) = 1 - 2 = -1.\]Step 5: State the conclusion.
The final result is \( -1 \).