If the functions f, g and h are defined from the set of real numbers R to R such that
`f(x)=x^(2)-1,g(x)=sqrt((x^(2)+1))`,
`h(x)={{:("0,","if",x<0),("x,","if",xge0):}`
Then find the composite function ho(fog)(x).

Explanation

To find the composite function \( h \circ (f \circ g)(x) \), we will follow these steps:

### Step 1: Find \( g(x) \)
Given:
\[ g(x) = \sqrt{x^2 + 1} \]
This function takes any real number \( x \) and returns the square root of \( x^2 + 1 \).

### Step 2: Find \( f(g(x)) \)
Now, we need to substitute \( g(x) \) into \( f(x) \):
\[ f(x) = x^2 - 1 \]
So,
\[ f(g(x)) = f(\sqrt{x^2 + 1}) = (\sqrt{x^2 + 1})^2 - 1 \]
Calculating this gives:
\[ f(g(x)) = x^2 + 1 - 1 = x^2 \]

### Step 3: Find \( h(f(g(x))) \)
Now we substitute \( f(g(x)) \) into \( h(x) \):
\[ h(x) = \begin{cases} 0 & \text{if } x < 0 \\ x & \text{if } x \geq 0 \end{cases} \]
Since \( f(g(x)) = x^2 \), we need to evaluate \( h(x^2) \).

### Step 4: Determine the value of \( h(x^2) \)
For any real number \( x \), \( x^2 \) is always greater than or equal to 0. Therefore:
\[ h(x^2) = x^2 \quad \text{(since \( x^2 \geq 0 \))} \]

### Final Result
Thus, the composite function \( h \circ (f \circ g)(x) \) is:
\[ h \circ (f \circ g)(x) = x^2 \]