If `f(a, b)=(a+b)/(2)`
`g(a, b)=a^(2)+b^(2)`
`h(a, b)="max "(a, b)`
Find the value of `f(g(3, 9)),h(-1, 1)`

Answer

To solve the problem, we need to evaluate the functions step by step.

### Step 1: Calculate \( g(3, 9) \)
The function \( g(a, b) \) is defined as:
\[ g(a, b) = a^2 + b^2 \]
Substituting \( a = 3 \) and \( b = 9 \):
\[ g(3, 9) = 3^2 + 9^2 \]
Calculating the squares:
\[ 3^2 = 9 \quad \text{and} \quad 9^2 = 81 \]
Now, adding these values:
\[ g(3, 9) = 9 + 81 = 90 \]

### Step 2: Calculate \( h(-1, 1) \)
The function \( h(a, b) \) is defined as:
\[ h(a, b) = \text{max}(a, b) \]
Substituting \( a = -1 \) and \( b = 1 \):
\[ h(-1, 1) = \text{max}(-1, 1) \]
The maximum value between -1 and 1 is:
\[ h(-1, 1) = 1 \]

### Step 3: Calculate \( f(g(3, 9), h(-1, 1)) \)
Now we have \( g(3, 9) = 90 \) and \( h(-1, 1) = 1 \). We need to evaluate:
\[ f(90, 1) \]
The function \( f(a, b) \) is defined as:
\[ f(a, b) = \frac{a + b}{2} \]
Substituting \( a = 90 \) and \( b = 1 \):
\[ f(90, 1) = \frac{90 + 1}{2} \]
Calculating the sum:
\[ 90 + 1 = 91 \]
Now, dividing by 2:
\[ f(90, 1) = \frac{91}{2} = 45.5 \]

### Final Answer
The value of \( f(g(3, 9), h(-1, 1)) \) is:
\[ \boxed{45.5} \]