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)

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:** \( \boxed{2 \sqrt{3} + 2} \)