, Given 2 f () 6 9 x x x and g x x ø ù . Find and
simplify the following: 1.1 g f x ø ù 1.2 2 110 0 x x e
e (5) 1.3 lnø ù øln5ù 25 x e e (3) 1.4 ø ù ø ù ø
ù 8 8 8 log 7 log 4x log 5 (3) 1.5 sin2 x cos2 x
sin x 0 on [0, 2 ). (5) 1.6 x sinø70 ùcosø25 ù
cosø70 ùsinø25 ù (3) [23]
𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏 𝟏
𝟏. 𝟏 𝑭𝒊𝒏𝒅 (𝒈 ∘ 𝒇)(𝒙)(𝒈 ∘ 𝒇)(𝒙)
𝐺𝑖𝑣𝑒𝑛 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠:
𝑓(𝑥) = 𝑥2 + 6𝑥 + 9𝑓(𝑥) = 𝑥2 + 6𝑥 + 9𝑔(𝑥) = 𝑥𝑔(𝑥) = 𝑥
𝐶𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑚𝑒𝑎𝑛𝑠 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔 𝑓(𝑥)𝑓(𝑥) 𝑖𝑛𝑡𝑜 𝑔(𝑥)𝑔(𝑥):
(𝑔 ∘ 𝑓)(𝑥) = 𝑔(𝑓(𝑥)) = 𝑔(𝑥2 + 6𝑥 + 9) = 𝑥2 + 6𝑥 + 9(𝑔 ∘ 𝑓)(𝑥)
= 𝑔(𝑓(𝑥)) = 𝑔(𝑥2 + 6𝑥 + 9) = 𝑥2 + 6𝑥 + 9
𝑆𝑖𝑛𝑐𝑒 𝑥2 + 6𝑥 + 9 = (𝑥 + 3)2𝑥2 + 6𝑥 + 9 = (𝑥 + 3)2 , 𝑤𝑒 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦:
(𝑥 + 3)2 =∣ 𝑥 + 3 ∣ (𝑥 + 3)2 =∣ 𝑥 + 3 ∣
𝟏. 𝟐 𝑺𝒐𝒍𝒗𝒆 𝟐𝒆𝒙 − 𝟏 = 𝟏𝟎𝒆𝒙𝟐𝒆𝒙 − 𝟏 = 𝟏𝟎𝒆𝒙
𝐷𝑖𝑣𝑖𝑑𝑒 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 𝑏𝑦 𝑒𝑥𝑒𝑥:
2𝑒𝑥 − 1𝑒𝑥 = 10𝑒𝑥𝑒𝑥𝑒𝑥2𝑒𝑥 − 1 = 𝑒𝑥10𝑒𝑥2𝑒 − 1 = 102𝑒 − 1 = 10
𝑆𝑖𝑛𝑐𝑒 𝑒 − 1 = 1𝑒𝑒 − 1 = 𝑒1, 𝑟𝑒𝑤𝑟𝑖𝑡𝑒 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛:
2𝑒 = 10𝑒2 = 10
𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑦 𝑒𝑒:
simplify the following: 1.1 g f x ø ù 1.2 2 110 0 x x e
e (5) 1.3 lnø ù øln5ù 25 x e e (3) 1.4 ø ù ø ù ø
ù 8 8 8 log 7 log 4x log 5 (3) 1.5 sin2 x cos2 x
sin x 0 on [0, 2 ). (5) 1.6 x sinø70 ùcosø25 ù
cosø70 ùsinø25 ù (3) [23]
𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏 𝟏
𝟏. 𝟏 𝑭𝒊𝒏𝒅 (𝒈 ∘ 𝒇)(𝒙)(𝒈 ∘ 𝒇)(𝒙)
𝐺𝑖𝑣𝑒𝑛 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠:
𝑓(𝑥) = 𝑥2 + 6𝑥 + 9𝑓(𝑥) = 𝑥2 + 6𝑥 + 9𝑔(𝑥) = 𝑥𝑔(𝑥) = 𝑥
𝐶𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑚𝑒𝑎𝑛𝑠 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔 𝑓(𝑥)𝑓(𝑥) 𝑖𝑛𝑡𝑜 𝑔(𝑥)𝑔(𝑥):
(𝑔 ∘ 𝑓)(𝑥) = 𝑔(𝑓(𝑥)) = 𝑔(𝑥2 + 6𝑥 + 9) = 𝑥2 + 6𝑥 + 9(𝑔 ∘ 𝑓)(𝑥)
= 𝑔(𝑓(𝑥)) = 𝑔(𝑥2 + 6𝑥 + 9) = 𝑥2 + 6𝑥 + 9
𝑆𝑖𝑛𝑐𝑒 𝑥2 + 6𝑥 + 9 = (𝑥 + 3)2𝑥2 + 6𝑥 + 9 = (𝑥 + 3)2 , 𝑤𝑒 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦:
(𝑥 + 3)2 =∣ 𝑥 + 3 ∣ (𝑥 + 3)2 =∣ 𝑥 + 3 ∣
𝟏. 𝟐 𝑺𝒐𝒍𝒗𝒆 𝟐𝒆𝒙 − 𝟏 = 𝟏𝟎𝒆𝒙𝟐𝒆𝒙 − 𝟏 = 𝟏𝟎𝒆𝒙
𝐷𝑖𝑣𝑖𝑑𝑒 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 𝑏𝑦 𝑒𝑥𝑒𝑥:
2𝑒𝑥 − 1𝑒𝑥 = 10𝑒𝑥𝑒𝑥𝑒𝑥2𝑒𝑥 − 1 = 𝑒𝑥10𝑒𝑥2𝑒 − 1 = 102𝑒 − 1 = 10
𝑆𝑖𝑛𝑐𝑒 𝑒 − 1 = 1𝑒𝑒 − 1 = 𝑒1, 𝑟𝑒𝑤𝑟𝑖𝑡𝑒 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛:
2𝑒 = 10𝑒2 = 10
𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑦 𝑒𝑒:
