3. Find a linear function \( f(x)=a x+b \) given that: a) \( f(2 x+3) \equiv-2 \) b) \( f(1-x) \equiv 4 x+1 \) c) \( f(f(x)) \equiv 4 x+3 \)

To find the linear function \( f(x) = ax + b \), we need to solve for \( a \) and \( b \) using the given conditions. a) \( f(2x + 3) = -2 \) Substitute \( 2x + 3 \) for \( x \) in the function \( f(x) \): \[ f(2x + 3) = a(2x + 3) + b \] \[ -2 = 2ax + 3a + b \] b) \( f(1 - x) = 4x + 1 \) Substitute \( 1 - x \) for \( x \) in the function \( f(x) \): \[ f(1 - x) = a(1 - x) + b \] \[ 4x + 1 = a - ax + b \] c) \( f(f(x)) = 4x + 3 \) First, we need to find \( f(x) \) and then substitute it into itself: \[ f(x) = ax + b \] \[ f(f(x)) = f(ax + b) = a(ax + b) + b = a^2x + ab + b \] \[ 4x + 3 = a^2x + ab + b \] Now we have three equations: 1. \( -2 = 2ax + 3a + b \) 2. \( 4x + 1 = a - ax + b \) 3. \( 4x + 3 = a^2x + ab + b \) Let's solve these equations step by step. From equation 1: \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a + b \] \[ -2 = 2ax + 3a