`f:R to R` given by f(x)=5-3 sin x, is

Options

A. one-one
B. onto
C. one-one and onto
D. none of these

Answer

To determine the nature of the function \( f: \mathbb{R} \to \mathbb{R} \) given by \( f(x) = 5 - 3 \sin x \), we will analyze its properties step by step.

### Step 1: Identify the basic function
The function is defined as:
\[ f(x) = 5 - 3 \sin x \]
Here, the core part of the function is \( \sin x \).

**Hint:** Start by understanding the basic sine function and its properties.

### Step 2: Analyze the sine function
The sine function, \( \sin x \), oscillates between -1 and 1 for all real numbers \( x \). Therefore, the range of \( \sin x \) is:
\[ [-1, 1] \]

**Hint:** Recall the range of the sine function and how it behaves.

### Step 3: Transform the sine function
Now, we need to consider the transformation applied to \( \sin x \). The function \( -3 \sin x \) will stretch the sine function by a factor of 3 and reflect it across the x-axis. Thus, the range of \( -3 \sin x \) will be:
\[ [-3, 3] \]

**Hint:** Understand how multiplying by a negative number affects the graph of a function.

### Step 4: Shift the function
Next, we add 5 to the transformed sine function:
\[ f(x) = 5 - 3 \sin x \]
This shifts the entire graph of \( -3 \sin x \) upward by 5 units. Therefore, the new range of \( f(x) \) becomes:
\[ [-3 + 5, 3 + 5] = [2, 8] \]

**Hint:** Remember that adding a constant shifts the graph vertically.

### Step 5: Determine the range and codomain
The range of \( f(x) \) is \( [2, 8] \), while the codomain is \( \mathbb{R} \), which includes all real numbers. Since the range is not equal to the codomain, we conclude that the function is not onto.

**Hint:** Compare the range of the function with its codomain to determine if it is onto.

### Step 6: Check for one-to-one or many-to-one
To check if the function is one-to-one (injective), we can observe that the sine function is periodic and takes the same value for different inputs. Therefore, \( f(x) \) will also yield the same output for multiple inputs. This indicates that the function is many-to-one.

**Hint:** Consider the periodic nature of the sine function when determining injectivity.

### Conclusion
Based on the analysis:
- The function \( f(x) = 5 - 3 \sin x \) is **many-to-one** because it maps multiple values of \( x \) to the same value of \( f(x) \).
- The function is **not onto** since its range \( [2, 8] \) does not cover all real numbers.

Thus, the final answer is that the function is many-one and into.

**Final Answer:** The function \( f(x) = 5 - 3 \sin x \) is many-one and into.