Let `f(x)=[x]` and `g(x)=|x|, AA x epsilon R` then value of `gof((-5)/3)+fog((-5)/3)` is equal to : (where `fog(x)=f(g(x)))`

Options

A. 1
B. `-1`
C. `-2`
D. `3`

Explanation

To solve the problem, we need to evaluate the expression \( g(f(-\frac{5}{3})) + f(g(-\frac{5}{3})) \) where \( f(x) = [x] \) (the greatest integer function) and \( g(x) = |x| \) (the absolute value function).

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

The greatest integer function \( f(x) = [x] \) gives the largest integer less than or equal to \( x \).

\[ -\frac{5}{3} \approx -1.6667 \]

The greatest integer less than or equal to \(-\frac{5}{3}\) is \(-2\).

\[ f(-\frac{5}{3}) = -2 \]

**Hint:** To find the greatest integer less than or equal to a number, visualize the number on a number line and identify the nearest integer to the left.

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

Now we apply the function \( g(x) = |x| \):

\[ g(-2) = |-2| = 2 \]

**Hint:** The absolute value function converts any negative number to its positive counterpart.

### Step 3: Calculate \( g(-\frac{5}{3}) \)

Next, we calculate \( g(-\frac{5}{3}) \):

\[ g(-\frac{5}{3}) = |-\frac{5}{3}| = \frac{5}{3} \]

**Hint:** Again, apply the absolute value function to convert the negative input to a positive output.

### Step 4: Calculate \( f(g(-\frac{5}{3})) = f(\frac{5}{3}) \)

Now we need to find \( f(\frac{5}{3}) \):

\[ \frac{5}{3} \approx 1.6667 \]

The greatest integer less than or equal to \(\frac{5}{3}\) is \(1\).

\[ f(\frac{5}{3}) = 1 \]

**Hint:** Use the same method as before to find the greatest integer less than or equal to a positive number.

### Step 5: Combine the results

Now we can combine the results from Steps 2 and 4:

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

### Final Answer

The value of \( g(f(-\frac{5}{3})) + f(g(-\frac{5}{3})) \) is \( \boxed{3} \).

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