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Question

If f(x) = 4x − x2, x ∈ R, then write the value of f(a + 1) −f(a − 1).

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Solution

Given:
f(x) = 4x − x2, x ∈ R
Now,
f(a + 1) = 4(a + 1) - (a + 1)2
= 4a + 4 - (a2 + 1 + 2a)
= 4a + 4 - a2 - 1 - 2a
= 2a - a2 + 3
f(a - 1) = 4(a - 1) - (a - 1)2
= 4a - 4 - (a2 + 1 - 2a)
= 4a - 4 - a2 - 1 + 2a
= 6a - a2 - 5
Thus,
f(a + 1) − f(a − 1) = ( 2a - a2 + 3) - (6a - a2 - 5)
= 2a - a2 + 3 - 6a + a2 + 5
= 8 - 4a
= 4(2 - a)

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