Chapter 1, Problem 8RE

To determine

To find: The domain and the range of the function.

Answer to Problem 8RE

Solution:

The domain of the function F(t) is $\left(-\infty ,\infty \right)$.

The range of the function F(t) is $\left[2,4\right]$.

Explanation of Solution

Given:

The function $F\left(t\right)=3+\cos 2t$.

Definition used:

The domain is the set of all input values of the function for which the function is real and defined.

Calculation:

Recall the fact that the domain of the cosine function is $\left(-\infty ,\infty \right)$ and the range $\left[-1,1\right]$.

Note that the adding 3 to the cosine function does not change the domain.

Thus, domain of the function $F\left(t\right)=3+\cos 2t$ is $\left(-\infty ,\infty \right)$.

The range of the cosine function is $\left[-1,1\right]$.

That is, $-1\le \cos 2t\le 1$.

For $3+\cos 2t$, the range of the function is obtained as follows.

$\begin{array}{l}-1+3\le \cos 2t+3\le 1+3\\ 2\le \cos 2t+3\le 4\end{array}$

Thus, it can be concluded that the range of the function is $\left[2,4\right]$.