Let f(x) be the greatest integer function and g(x) be the modulus function.
What is `(gof)(-(5)/(3))-(fog)(-(5)/(3))` equal to?

Options

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

Explanation

To solve the problem, we need to evaluate the expression \( (g \circ f)\left(-\frac{5}{3}\right) - (f \circ g)\left(-\frac{5}{3}\right) \) where \( f(x) \) is the greatest integer function (also known as the floor function) and \( g(x) \) is the modulus function.

### Step 1: Determine \( g(f(-\frac{5}{3})) \)

1. **Calculate \( f(-\frac{5}{3}) \)**:
- The greatest integer function \( f(x) \) gives the greatest integer less than or equal to \( x \).
- For \( -\frac{5}{3} \approx -1.67 \), the greatest integer less than or equal to \( -1.67 \) is \( -2 \).
- Therefore, \( f(-\frac{5}{3}) = -2 \).

2. **Calculate \( g(f(-\frac{5}{3})) = g(-2) \)**:
- The modulus function \( g(x) \) gives the absolute value of \( x \).
- So, \( g(-2) = |-2| = 2 \).

Thus, \( g(f(-\frac{5}{3})) = 2 \).

### Step 2: Determine \( f(g(-\frac{5}{3})) \)

1. **Calculate \( g(-\frac{5}{3}) \)**:
- Using the modulus function, \( g(-\frac{5}{3}) = |-\frac{5}{3}| = \frac{5}{3} \).

2. **Calculate \( f(g(-\frac{5}{3})) = f(\frac{5}{3}) \)**:
- For \( \frac{5}{3} \approx 1.67 \), the greatest integer less than or equal to \( 1.67 \) is \( 1 \).
- Therefore, \( f(\frac{5}{3}) = 1 \).

Thus, \( f(g(-\frac{5}{3})) = 1 \).

### Step 3: Combine the results

Now we can substitute back into the original expression:

\[ (g \circ f)\left(-\frac{5}{3}\right) - (f \circ g)\left(-\frac{5}{3}\right) = 2 - 1 = 1 \]

### Final Answer

The value of \( (g \circ f)\left(-\frac{5}{3}\right) - (f \circ g)\left(-\frac{5}{3}\right) \) is \( 1 \).

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