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: {Compute \( f(-8/5) \)}
\[ f \left( \frac{-8}{5} \right) = \left\lfloor \frac{-8}{5} \right\rfloor = -2. \] Step 2: {Compute \( g(8/5) \) 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: {Compute \( f(g(8/5)) \) 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: {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: {Conclusion}
Thus, the correct answer is \( -1 \).

Answer 2

Step 1: Evaluate \( g\left( \frac{8}{5} \right) \)
We are given that \( g(x) = |x| \). So, \[ g\left( \frac{8}{5} \right) = \left| \frac{8}{5} \right| = \frac{8}{5}. \] Step 2: Evaluate \( f\left( g\left( \frac{8}{5} \right) \right) \)
Now, we need to evaluate \( f\left( g\left( \frac{8}{5} \right) \right) \), which is \( f\left( \frac{8}{5} \right) \). Recall that \( f(x) = [x] \), where \( [x] \) denotes the greatest integer less than or equal to \( x \). Thus, \[ f\left( \frac{8}{5} \right) = \left\lfloor \frac{8}{5} \right\rfloor = \left\lfloor 1.6 \right\rfloor = 1. \] Step 3: Evaluate \( f\left( \frac{-8}{5} \right) \)
Next, we evaluate \( f\left( \frac{-8}{5} \right) \). Using the definition of \( f(x) = [x] \), \[ f\left( \frac{-8}{5} \right) = \left\lfloor \frac{-8}{5} \right\rfloor = \left\lfloor -1.6 \right\rfloor = -2. \] Step 4: Evaluate \( g\left( f\left( \frac{-8}{5} \right) \right) \)
Now, we need to evaluate \( g\left( f\left( \frac{-8}{5} \right) \right) \), which is \( g(-2) \). Since \( g(x) = |x| \), \[ g(-2) = | -2 | = 2. \] Step 5: Combine the results
We are asked to find the value of: \[ f\left( g\left( \frac{8}{5} \right) \right) - g\left( f\left( \frac{-8}{5} \right) \right). \] Substituting the values we have computed: \[ f\left( g\left( \frac{8}{5} \right) \right) = 1 \quad \text{and} \quad g\left( f\left( \frac{-8}{5} \right) \right) = 2. \] Thus, \[ f\left( g\left( \frac{8}{5} \right) \right) - g\left( f\left( \frac{-8}{5} \right) \right) = 1 - 2 = -1. \] Final Answer:
The value is:

-1