A mapping f: R `rarr` R which is defined as f(x) = cos x: x `in` R is:

Options

A. Only one-one
B. Only onto
C. One-one onto
D. Neither one-one nor onto

Answer

To determine the nature of the function \( f(x) = \cos x \) where \( x \in \mathbb{R} \), we need to check if it is a one-one function and if it is an onto function.

### Step 1: Check if \( f(x) \) is a one-one function

A function is one-one (injective) if it never assigns the same value to two different domain elements. In other words, for \( f(x_1) = f(x_2) \), it must follow that \( x_1 = x_2 \).

1. **Graph the function**: The graph of \( f(x) = \cos x \) is a wave-like function that oscillates between -1 and 1.
2. **Draw horizontal lines**: If we draw horizontal lines (lines parallel to the x-axis) across the graph, we can see that each horizontal line intersects the graph at multiple points. For example, the line \( y = 0.5 \) intersects the cosine curve at multiple values of \( x \).
3. **Conclusion**: Since horizontal lines intersect the graph at more than one point, \( f(x) = \cos x \) is not a one-one function. It is classified as a many-one function.

### Step 2: Check if \( f(x) \) is an onto function

A function is onto (surjective) if every element in the codomain is mapped by at least one element in the domain.

1. **Identify the range of \( f(x) \)**: The range of \( f(x) = \cos x \) is the set of all possible output values, which is \([-1, 1]\).
2. **Identify the codomain**: The codomain given is \( \mathbb{R} \) (the set of all real numbers).
3. **Compare range and codomain**: The range \([-1, 1]\) is not equal to the codomain \( \mathbb{R} \) because the codomain includes all real numbers, while the range is limited to values between -1 and 1.
4. **Conclusion**: Since not every element in the codomain \( \mathbb{R} \) is covered by the range of \( f(x) \), the function is not onto.

### Final Conclusion

Based on the analysis:
- The function \( f(x) = \cos x \) is neither one-one nor onto.

Thus, the answer is that the function is neither one-one nor onto.

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