31) A function is defined for all positive numbers $$ x $$ as $$ f(x)=a \sqrt{x}+b $$. What is the value of $$ f(3) $$, if f(4) $$ -f(1)=2 $$ and $$ f(4)+f(1)=10 $$ ? (A) 1 (B) 2 (C) $$ 2 \sqrt{3} $$ (D) $$ 2 \sqrt{3}+2 $$ 32)

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To determine the value of $$ f(3) $$ for the function $$ f(x) = a \sqrt{x} + b $$, we'll use the given conditions:

1. $$ f(4) - f(1) = 2 $$
2. $$ f(4) + f(1) = 10 $$

**Step 1: Express $$ f(4) $$ and $$ f(1) $$ in terms of $$ a $$ and $$ b $$**

$$
f(4) = a \sqrt{4} + b = 2a + b
$$
$$
f(1) = a \sqrt{1} + b = a + b
$$

**Step 2: Use the first condition to find $$ a $$**

$$
f(4) - f(1) = (2a + b) - (a + b) = a = 2
$$

So, $$ a = 2 $$.

**Step 3: Use the second condition to find $$ b $$**

$$
f(4) + f(1) = (2a + b) + (a + b) = 3a + 2b = 10
$$

Substitute $$ a = 2 $$:

$$
3(2) + 2b = 10 \implies 6 + 2b = 10 \implies 2b = 4 \implies b = 2
$$

**Step 4: Determine $$ f(3) $$**

$$
f(3) = a \sqrt{3} + b = 2\sqrt{3} + 2
$$

**Answer:** $$ \boxed{2 \sqrt{3} + 2} $$
$$ f(3) = 2\sqrt{3} + 2 $$

31) A function is defined for all positive numbers \( x \) as \( f(x)=a \sqrt{x}+b \). What is the value of \( f(3) \), if f(4) \( -f(1)=2 \) and \( f(4)+f(1)=10 \) ? (A) 1 (B) 2 (C) \( 2 \sqrt{3} \) (D) \( 2 \sqrt{3}+2 \) 32)

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