If f: `R rarr S` defined by f(x) = 4 sin x-3 cos x 1 is onto, then what is S equal to?

Options

A. (-5.5)
B. (-5.5)
C. `(-4,6)`
D. `(-4,6)`

Answer

To solve the problem, we need to find the range of the function \( f(x) = 4 \sin x - 3 \cos x + 1 \) and determine what the set \( S \) is, given that the function is onto.

### Step-by-Step Solution:

1. **Identify the Function**:
We have the function \( f(x) = 4 \sin x - 3 \cos x + 1 \).

2. **Rewrite the Function**:
The function can be expressed in the form \( A \sin(x) + B \cos(x) + C \), where \( A = 4 \), \( B = -3 \), and \( C = 1 \).

3. **Find Maximum and Minimum Values**:
To find the range, we first calculate the maximum and minimum values of the expression \( 4 \sin x - 3 \cos x \).

- The maximum value of \( A \sin(x) + B \cos(x) \) can be derived using the formula:
\[ R = \sqrt{A^2 + B^2} \]
where \( R \) is the amplitude.

- Here, \( A = 4 \) and \( B = -3 \):
\[ R = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

4. **Calculate Maximum and Minimum Values**:
- The maximum value of \( f(x) \) is:
\[ \text{Max} = R + C = 5 + 1 = 6 \]
- The minimum value of \( f(x) \) is:
\[ \text{Min} = -R + C = -5 + 1 = -4 \]

5. **Determine the Range**:
The range of the function \( f(x) \) is from the minimum value to the maximum value:
\[ \text{Range} = (-4, 6) \]

6. **Conclusion**:
Since the function is onto, the set \( S \) is equal to the range of the function:
\[ S = (-4, 6) \]

### Final Answer:
The set \( S \) is equal to \( (-4, 6) \).

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