(a.) 5+cosx
Since, −1≤cosx≤1
Adding 5 on each sides.
⇒−1+5≤5+cosx≤1+5
⇒4≤5+cosx≤6
Hence, 5+cosx∈[4,6]
(b.) 0.5cosx−4
Since, −1≤cosx≤1
Multiplying 0.5 on each sides.
⇒−0.5≤0.5cosx≤0.5
Now, subtracting 4 from each sides, we get:
⇒−0.5−4≤0.5cosx−4≤0.5−4
⇒−4.5≤0.5cosx−4≤−3.5
Hence, 0.5cosx−4∈[−4.5,−3.5]
(c.) 7−3cosx
Since, −1≤cosx≤1
Multiplying with −3 on each sides, we get:
3≥−3cosx≥−3
Now, adding 7 we get:
7+3≥7−3cosx≥−3+7
⇒10≥7−3cosx≥4
Hence, 7−3cosx∈[4,10]
(d.) −14cosx
Since, −1≤cosx≤1
Multiplying with −14 on each sides, we get:
−14×−1≥−14cosx≥1×−14
14≥−14cosx≥−14
Hence, −14sinx∈[−14,14]