21. If a function \( f \) is defines as \( f(x+2)=f(x)+f(2), x \longleftarrow-R \) Se defines by \( f(x)=2 x+3 \), then find \( f(0), f(1) \),\n i. \( f(0)=0 \) ii. \( f(-2)=-f(2) \)
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To find \( f(0) \) and \( f(1) \), we start by observing the given function \( f(x) = 2x + 3 \). First, we calculate \( f(0) \): \[ f(0) = 2(0) + 3 = 3. \] Next, we calculate \( f(1) \): \[ f(1) = 2(1) + 3 = 5. \] Now, regarding the additional statements: i. False, because \( f(0) = 3 \). ii. Let's verify \( f(-2) \) and \( f(2) \): \[ f(-2) = 2(-2) + 3 = -4 + 3 = -1, \] \[ f(2) = 2(2) + 3 = 4 + 3 = 7. \] So, \( -f(2) = -7 \), which means that \( f(-2) = -1 \) is not equal to \( -f(2) \). Both statements are incorrect based on the calculations.