The number of elements in the range of functions: `y=sin^(-1) [x^(2)+5/9]+cos^(-1) [x^(2)-4/9]` where where [.] denotes the greatest integer function is:

Answer

To find the number of elements in the range of the function \( y = \sin^{-1} \left( \lfloor x^2 + \frac{5}{9} \rfloor \right) + \cos^{-1} \left( \lfloor x^2 - \frac{4}{9} \rfloor \right) \), we will follow these steps:

### Step 1: Determine the range of \( x^2 \)

The expression \( x^2 \) can take any non-negative value, meaning \( x^2 \geq 0 \). However, we need to find the conditions under which the arguments of the inverse sine and cosine functions are valid.

### Step 2: Find the conditions for \( \lfloor x^2 + \frac{5}{9} \rfloor \)

For \( \sin^{-1} \left( \lfloor x^2 + \frac{5}{9} \rfloor \right) \) to be defined, \( \lfloor x^2 + \frac{5}{9} \rfloor \) must be in the range \([-1, 1]\).

1. **Lower Bound**:
\[ \lfloor x^2 + \frac{5}{9} \rfloor \geq -1 \implies x^2 + \frac{5}{9} \geq -1 \implies x^2 \geq -1 - \frac{5}{9} \implies x^2 \geq -\frac{14}{9} \text{ (always true since } x^2 \geq 0\text{)} \]

2. **Upper Bound**:
\[ \lfloor x^2 + \frac{5}{9} \rfloor < 2 \implies x^2 + \frac{5}{9} < 2 \implies x^2 < 2 - \frac{5}{9} \implies x^2 < \frac{13}{9} \]

Thus, we have:
\[ 0 \leq x^2 < \frac{13}{9} \]

### Step 3: Find the conditions for \( \lfloor x^2 - \frac{4}{9} \rfloor \)

For \( \cos^{-1} \left( \lfloor x^2 - \frac{4}{9} \rfloor \right) \) to be defined, \( \lfloor x^2 - \frac{4}{9} \rfloor \) must also be in the range \([-1, 1]\).

1. **Lower Bound**:
\[ \lfloor x^2 - \frac{4}{9} \rfloor \geq -1 \implies x^2 - \frac{4}{9} \geq -1 \implies x^2 \geq -1 + \frac{4}{9} \implies x^2 \geq -\frac{5}{9} \text{ (always true)} \]

2. **Upper Bound**:
\[ \lfloor x^2 - \frac{4}{9} \rfloor < 2 \implies x^2 - \frac{4}{9} < 2 \implies x^2 < 2 + \frac{4}{9} \implies x^2 < \frac{22}{9} \]

Thus, we have:
\[ 0 \leq x^2 < \frac{22}{9} \]

### Step 4: Combine the conditions

The conditions we derived are:
1. \( 0 \leq x^2 < \frac{13}{9} \)
2. \( 0 \leq x^2 < \frac{22}{9} \)

The more restrictive condition is:
\[ 0 \leq x^2 < \frac{13}{9} \]

### Step 5: Evaluate the range of \( y \)

Now we need to evaluate \( y \) at the endpoints of the range of \( x^2 \):
- When \( x^2 = 0 \):
\[ y = \sin^{-1} \left( \lfloor 0 + \frac{5}{9} \rfloor \right) + \cos^{-1} \left( \lfloor 0 - \frac{4}{9} \rfloor \right) = \sin^{-1}(0) + \cos^{-1}(-1) = 0 + \pi = \pi \]

- When \( x^2 \) approaches \( \frac{13}{9} \):
\[ \lfloor \frac{13}{9} + \frac{5}{9} \rfloor = \lfloor \frac{18}{9} \rfloor = 2 \text{ (not valid)} \]
Thus, we check values just below \( \frac{13}{9} \).

For \( x^2 = 1 \):
\[ y = \sin^{-1} \left( \lfloor 1 + \frac{5}{9} \rfloor \right) + \cos^{-1} \left( \lfloor 1 - \frac{4}{9} \rfloor \right) = \sin^{-1}(1) + \cos^{-1}(0) = \frac{\pi}{2} + \frac{\pi}{2} = \pi \]

### Conclusion

Since both evaluations lead to \( y = \pi \) and there are no other values of \( y \) that we can derive from the given conditions, the number of elements in the range of the function is:

\[ \boxed{1} \]