Let's address the two problems in the image step by step.

Problem 21:
The function is given as: $f(x) = -30 \sin(2x) + 15$

We need to find the maximum and minimum values of $f(x)$.

Solution:

• The sine function $\sin(2x)$ oscillates between -1 and 1, so the range of $\sin(2x)$ is: $-1 \leq \sin(2x) \leq 1$

• Multiply this range by -30: $-30 \times (-1) \leq -30 \sin(2x) \leq -30 \times 1$ $30 \leq -30 \sin(2x) \leq -30$

• Now add 15 to the entire inequality: $30 + 15 \leq -30 \sin(2x) + 15 \leq -30 + 15$ $45 \leq f(x) \leq -15$

Thus, the maximum value is $45$ and the minimum value is $-15$.

The correct answer is B) max = 45, min = -15.

Problem 22:
We need to identify which of the given statements is true.

Let's examine each option:

A) The period of $f(x) = \tan(2\pi x + 1)$ is $\frac{1}{2}$.

• The period of the tangent function $\tan(kx)$ is $\frac{\pi}{k}$. Here, $k = 2\pi$, so the period is: $\frac{\pi}{2\pi} = \frac{1}{2}$ This statement is True.

B) The amplitude of $f(x) = 3 \tan(2x + 3)$ is 3.

• The tangent function does not have an amplitude because it has vertical asymptotes and ranges from $-\infty$ to $\infty$. Therefore, this statement is False.

C) The function $f(x) = -\frac{1}{2} \sin(2x - 4) + 1$ is shifted vertically downwards.

• The function $f(x) = -\frac{1}{2} \sin(2x - 4) + 1$ is shifted vertically upwards by 1 unit, not downwards. Therefore, this statement is False.

D) The function $f(x) = 3 \cos(0.5x + 3) + 2$ is shifted horizontally to the right.

• The function $f(x) = 3 \cos(0.5x + 3) + 2$ has a phase shift. To find the phase shift, solve $0.5x + 3 = 0$: $x = -\frac{6}{0.5} = -6$ This indicates a shift to the left by 6 units, not to the right. Therefore, this statement is False.

Thus, the correct answer is A) The period of $f(x) = \tan(2\pi x + 1)$ is $\frac{1}{2}$.

Would you like more details or have any other questions? Here are some related questions for practice:

1. How do you find the period of a sine or cosine function?
2. What is the range of a general sine function $f(x) = A \sin(Bx) + C$?
3. How does the amplitude of a cosine function affect its graph?
4. What are the key transformations for the graph of a tangent function?
5. Can you describe how horizontal phase shifts affect the graph of trigonometric functions?

Tip: Always remember that the amplitude only applies to sine and cosine functions, while tangent and cotangent have no amplitude.