Chapter 4, Problem 4PS

a.

To determine

Find if the statement is true or false.

Answer to Problem 4PS

  $\boxed{True}$

Explanation of Solution

Given information:

The function $f$ is periodic, with period c. So, $f\left(t+c\right)=f\left(t\right)$ . Determine whether each statement is true or false. Explain.

  $f\left(t-2c\right)=f\left(t\right)$

Calculation:

Here, we will consider the following function:

  $f\left(t-2c\right)=f\left(t\right)$

Hence, the given statement is $\boxed{True}$ because $2c$ is two periods.

b.

To determine

Find if the statement is true or false.

Answer to Problem 4PS

  $\boxed{False}$

Explanation of Solution

Given information:

The function $f$ is periodic, with period c. So, $f\left(t+c\right)=f\left(t\right)$ . Determine whether each statement is true or false. Explain.

  $f\left(t+\frac{1}{2}c\right)=f\left(\frac{1}{2}t\right)$

Calculation:

Here, we will consider the following function:

  $f\left(t+\frac{1}{2}c\right)=f\left(\frac{1}{2}t\right)$

Hence, the given statement is $\boxed{False}$ because $\frac{c}{2}$ is half the period of the function on the left and the function on the right has twice the period of the function on the left.