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 𝒂𝒏𝒅 𝒔𝒊𝒎𝒑𝒍𝒊𝒇𝒚 𝒈(𝒇(𝒙))𝒈(𝒇(𝒙))

𝑊𝑒 𝑎𝑟𝑒 𝑔𝑖𝑣𝑒𝑛:

𝑓(𝑥) = 2𝑥2 + 6𝑥 + 9𝑓(𝑥) = 2𝑥2 + 6𝑥 + 9
𝑎𝑛𝑑

𝑔(𝑥) = 𝑥𝑔(𝑥) = 𝑥

𝑇𝑜 𝑓𝑖𝑛𝑑 𝑔(𝑓(𝑥))𝑔(𝑓(𝑥)), 𝑤𝑒 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑓(𝑥)𝑓(𝑥) 𝑖𝑛𝑡𝑜 𝑔(𝑥)𝑔(𝑥):

𝑔(𝑓(𝑥)) = 𝑔(2𝑥2 + 6𝑥 + 9) = 2𝑥2 + 6𝑥 + 9𝑔(𝑓(𝑥)) = 𝑔(2𝑥2 + 6𝑥 + 9)
= 2𝑥2 + 6𝑥 + 9
𝑇ℎ𝑢𝑠, 𝑡ℎ𝑒 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑖𝑒𝑑 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑖𝑠:

𝑔(𝑓(𝑥)) = 2𝑥2 + 6𝑥 + 9𝑔(𝑓(𝑥)) = 2𝑥2 + 6𝑥 + 9



𝟏. 𝟐 𝑺𝒐𝒍𝒗𝒆 𝟐𝒆𝒙 − 𝟏 − 𝒆𝒙 − 𝟐 = 𝟎𝟐𝒆𝒙 − 𝟏 − 𝒆𝒙 − 𝟐 = 𝟎

𝐹𝑖𝑟𝑠𝑡, 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑢𝑡 𝑡ℎ𝑒 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 𝑡𝑒𝑟𝑚𝑠:

𝑒𝑥 − 2(2𝑒 − 1) = 0𝑒𝑥 − 2(2𝑒 − 1) = 0

𝑆𝑖𝑛𝑐𝑒 𝑒𝑥 − 2 ≠ 0𝑒𝑥 − 2 = 0 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑥𝑥, 𝑤𝑒 𝑠𝑜𝑙𝑣𝑒: