Functions and Graphs

22.
(a)
f(x) = x² - kx

f(x + 2)
= (x + 2)² - k(x + 2)
= x² + 4x + 4 - kx - 2k
= x² + (4 - k)x + 4 - 2k

f(x - 2)
= (x - 2)² - k(x - 2)
= x² - 4x + 4 - kx + 2k
= x² - (4 + k)x + 4 + 2k


(b)
f(x + 2) - f(x - 2) = kx - 32
(x² + 4x + 4 - kx - 2k) - (x² - 4x + 4 - kx + 2k) = kx - 32
x² + 4x + 4 - kx - 2k - x² + 4x - 4 + kx - 2k = kx - 32
8x - 4k = kx - 32

Compare the x terms on the both sides :
k = 8

Compare the constant terms on the both sides :
-4k = -32
k = 8

2011-11-11 19:14:25 補充：
再計算下去是要驗算題目有沒有出錯。因為若果兩個計出來 k 的數值不相同，則無解。

2011-11-11 19:19:11 補充：
8x - 4k = kx - 32

Compare the x terms on the both sides :
x term on the left hand side = 8x
x term on the right hand side = kx
Hence, 8x = kx
k = 8

Why 4 ?

for 22b only.

as f(x+2) - f(x-2) = kx-32 is just an equation, not an identical equation, so we cannot compare the coefficients directly at both side to get value k.

from others, finally got 8x - kx = 4k - 32
8x + 32 = kx + 4k
8(x+4) = k(x+4)
=> k = 8