Question

what is the range of $y = \cos x$?\
\
a. $x \
eq n\pi$\
\
b. all real numbers\
\
c. $-1 \leq y \leq 1$\
\
d. $x \
eq \frac{\pi}{2} + n\pi$

Explanation:

Step1: Recall the definition of range

The range of a function is the set of all possible output values (y - values) it can take.

Step2: Analyze the cosine function

The cosine function \( y = \cos x \) is a periodic function. From the unit circle definition or the graph of the cosine function, we know that the value of \( \cos x \) oscillates between - 1 and 1, inclusive. That is, for any real number \( x \), \( - 1\leqslant\cos x\leqslant1 \), or \( - 1\leqslant y\leqslant1 \) when \( y = \cos x \).

Step3: Eliminate other options

• Option A: \( x

eq n\pi \) is related to the domain of some other functions (like \( y=\tan x \) has domain \( x
eq\frac{\pi}{2}+n\pi \), \( y = \csc x \) has domain \( x
eq n\pi \)), not the range of \( y = \cos x \).

• Option B: The cosine function does not take all real numbers as output, it is bounded between - 1 and 1.
• Option D: \( x

eq\frac{\pi}{2}+n\pi \) is the domain of \( y=\tan x \), not related to the range of \( y = \cos x \).

Answer:

C. \( - 1\leqslant y\leqslant1 \)