If f(x) =
Question:

If f(x) =  4x − x2x ∈ R, then write the value of f(a + 1) −f(a − 1).

Solution:

Given:

f(x) =  4x − x2x ∈ R

Now,

$f(a+1)=4(a+1)-(a+1)^{2}$

$=4 a+4-\left(a^{2}+1+2 a\right)$

$=4 a+4-a^{2}-1-2 a$

 

$=2 a-a^{2}+3$

$f(a-1)=4(a-1)-(a-1)^{2}$

$=4 a-4-\left(a^{2}+1-2 a\right)$

$=4 a-4-a^{2}-1+2 a$

 

$=6 a-a^{2}-5$

Thus,

$f(a+1)-f(a-1)=\left(2 a-a^{2}+3\right)-\left(6 a-a^{2}-5\right)$

$=2 a-a^{2}+3-6 a+a^{2}+5$

$=8-4 a$

 

$=4(2-a)$

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