Question

Match the following functions with their corresponding range.

Answer 1

(a.) $5+\cos x$
Since, $-1\le \cos x\le 1$
Adding $5$ on each sides.
$\Rightarrow -1+5\le 5+\cos x\le 1+5$
$\Rightarrow 4\le 5+\cos x\le 6$
Hence, $5+\cos x\in [4,6]$

(b.) $0.5\cos x-4$
Since, $-1\le \cos x\le 1$
Multiplying $0.5$ on each sides.
$\Rightarrow -0.5\le 0.5\cos x\le 0.5$
Now, subtracting $4$ from each sides, we get:
$\Rightarrow -0.5-4\le 0.5\cos x-4\le 0.5-4$
$\Rightarrow -4.5\le 0.5\cos x-4\le -3.5$
Hence, $0.5\cos x-4\in [-4.5,-3.5]$

(c.) $7-3\cos x$
Since, $-1\le \cos x\le 1$
Multiplying with $-3$ on each sides, we get:
$3\ge -3\cos x\ge -3$
Now, adding $7$ we get:
$7+3\ge 7-3\cos x\ge -3+7$
$\Rightarrow 10\ge 7-3\cos x\ge 4$
Hence, $7-3\cos x\in [4,10]$

(d.) $-14\cos x$
Since, $-1\le \cos x\le 1$
Multiplying with $-14$ on each sides, we get:
$-14\times -1\ge -14\cos x\ge 1\times -14$
$14\ge -14\cos x\ge -14$
Hence, $-14\sin x\in [-14,14]$