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]
𝟏. 𝟏 𝑮𝒊𝒗𝒆𝒏 𝒇(𝒙) = 𝒙𝟐 + 𝟔𝒙 + 𝟗𝒇(𝒙) = 𝒙𝟐 + 𝟔𝒙 + 𝟗 𝒂𝒏𝒅 𝒈(𝒙) = 𝒙𝒈(𝒙)
= 𝒙, 𝒇𝒊𝒏𝒅 𝒈(𝒇(𝒙))𝒈(𝒇(𝒙)).
𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏:

1. 𝑪𝒐𝒎𝒑𝒐𝒔𝒆 𝒕𝒉𝒆 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔:

𝑔(𝑓(𝑥)) = 𝑔(𝑥2 + 6𝑥 + 9) = 𝑥2 + 6𝑥 + 9𝑔(𝑓(𝑥)) = 𝑔(𝑥2 + 6𝑥 + 9)
= 𝑥2 + 6𝑥 + 9
2. 𝑺𝒊𝒎𝒑𝒍𝒊𝒇𝒚 𝒕𝒉𝒆 𝒆𝒙𝒑𝒓𝒆𝒔𝒔𝒊𝒐𝒏 𝒖𝒏𝒅𝒆𝒓 𝒕𝒉𝒆 𝒔𝒒𝒖𝒂𝒓𝒆 𝒓𝒐𝒐𝒕:
𝑥2 + 6𝑥 + 9 = (𝑥 + 3)2𝑥2 + 6𝑥 + 9 = (𝑥 + 3)2
3. 𝑻𝒂𝒌𝒆 𝒕𝒉𝒆 𝒔𝒒𝒖𝒂𝒓𝒆 𝒓𝒐𝒐𝒕:
(𝑥 + 3)2 =∣ 𝑥 + 3 ∣ (𝑥 + 3)2 =∣ 𝑥 + 3 ∣
𝑭𝒊𝒏𝒂𝒍 𝑨𝒏𝒔𝒘𝒆𝒓:

∣ 𝑥 + 3 ∣∣ 𝑥 + 3 ∣


𝟏. 𝟐 𝑺𝒐𝒍𝒗𝒆 𝟐𝒆𝒙 − 𝟏𝟎 = 𝟎𝟐𝒆𝒙 − 𝟏𝟎 = 𝟎.

𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏:

1. 𝑰𝒔𝒐𝒍𝒂𝒕𝒆 𝒕𝒉𝒆 𝒆𝒙𝒑𝒐𝒏𝒆𝒏𝒕𝒊𝒂𝒍 𝒕𝒆𝒓𝒎:

2𝑒𝑥 − 10 = 0 ⟹ 2𝑒𝑥 = 10 ⟹ 𝑒𝑥 = 52𝑒𝑥 − 10 = 0 ⟹ 2𝑒𝑥 = 10
⟹ 𝑒𝑥 = 5