Find \( f(a+1)-f(a-1) \)
Question:
If
\[ f(x)=4x-x^2,\qquad x\in R \]
then write the value of
\[ f(a+1)-f(a-1) \]
Solution:
\[ f(a+1) = 4(a+1)-(a+1)^2 \]
\[ =4a+4-a^2-2a-1 \]
\[ =-a^2+2a+3 \]
Also,
\[ f(a-1) = 4(a-1)-(a-1)^2 \]
\[ =4a-4-a^2+2a-1 \]
\[ =-a^2+6a-5 \]
Therefore,
\[ f(a+1)-f(a-1) \]
\[ =(-a^2+2a+3)-(-a^2+6a-5) \]
\[ =8-4a \]
Hence,
\[ \boxed{8-4a} \]