Consider the function f: R→R defined by f(x)=sin(x)+cos(2x). Which of the following statements about f(x) is true?

To determine which statement about the function $f(x) = \sin(x) + \cos(2x)$ is true, we can analyze the function by looking into its properties. Here are the steps to analyze the function:

1. Break Down the Problem

Identify the properties of the function $f(x)$:

• Domain and range
• Periodicity
• Critical points and behavior of the function

2. Relevant Concepts

• The function $f(x)$ is composed of $\sin(x)$ and $\cos(2x)$.
• The sine function $\sin(x)$ has a period of $2\pi$.
• The cosine function $\cos(2x)$ has a period of $\pi$.
• Properties of combination of periodic functions.

3. Analysis and Detail

• The individual function $\sin(x)$ oscillates between -1 and 1.
• The function $\cos(2x)$ also oscillates between -1 and 1 but does so twice as fast.
• To find the overall periodicity of $f(x)$, we take the least common multiple of the periods of $\sin(x)$ and $\cos(2x)$. The period of $f(x)$ is $2\pi$ (since $\cos(2x)$ will complete its cycles twice within one period of $\sin(x)$).
• The range of $f(x)$ can be assessed:
  • Maximum of $f(x) = 1 + 1 = 2$
  • Minimum of $f(x) = -1 - 1 = -2$
  • Therefore, the range of $f(x)$ is between -2 and 2.

4. Verify and Summarize

The overall properties of $f(x) = \sin(x) + \cos(2x)$:

• Domain: All real numbers $\mathbb{R}$
• Periodicity: $2\pi$
• Range: [-2, 2]

Final Answer

Thus, the statement that is true about $f(x)$ is that it is periodic with a period of $2\pi$ and its range is $[-2, 2]$.