MAT1514 Assignment 1 (COMPLETE ANSWERS) 2025

, 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: