Let `f:RrarrR` and `g:RrarrR` be two functions defined by `f(x)=|x|` and `g(x)=[x]`, where [x] denotes the greatest integer less than or equal to x. Find (fog) (5.75) and (gof) - `(-sqrt(5))`.

Answer

To solve the problem, we need to find the values of (fog)(5.75) and (gof)(-√5) for the functions \( f(x) = |x| \) and \( g(x) = [x] \), where \([x]\) denotes the greatest integer less than or equal to \( x \).

### Step 1: Calculate (fog)(5.75)

1. **Find g(5.75)**:
- The function \( g(x) = [x] \) gives the greatest integer less than or equal to \( x \).
- For \( x = 5.75 \), the greatest integer less than or equal to 5.75 is 5.
- Therefore, \( g(5.75) = 5 \).

2. **Find f(g(5.75)) = f(5)**:
- Now we need to find \( f(5) \) using the function \( f(x) = |x| \).
- Since \( f(5) = |5| = 5 \).
- Thus, \( (fog)(5.75) = f(g(5.75)) = f(5) = 5 \).

### Step 2: Calculate (gof)(-√5)

1. **Find f(-√5)**:
- We first need to calculate \( f(-\sqrt{5}) \) using \( f(x) = |x| \).
- Since \( f(-\sqrt{5}) = |-\sqrt{5}| = \sqrt{5} \).

2. **Find g(f(-√5)) = g(√5)**:
- Now we need to find \( g(\sqrt{5}) \).
- The value of \( \sqrt{5} \) is approximately 2.236.
- The greatest integer less than or equal to \( \sqrt{5} \) is 2.
- Therefore, \( g(\sqrt{5}) = [\sqrt{5}] = 2 \).

3. **Final Result**:
- Thus, \( (gof)(-\sqrt{5}) = g(f(-\sqrt{5})) = g(\sqrt{5}) = 2 \).

### Summary of Results:
- \( (fog)(5.75) = 5 \)
- \( (gof)(-\sqrt{5}) = 2 \)