Let `R rarr R` be a function defined as f(x) = |x| for each `x in R` . - being the set of real numbers. Which ne of the following is correct?

Options

A. A. f is onto but not onto
B. B. f is onto but not one-one
C. C. f is both one-one and onto
D. D. f is neither one-one not tonto

Answer

To determine the properties of the function \( f(x) = |x| \), we need to check if it is one-to-one (injective) and onto (surjective).

### Step 1: Check if the function is one-to-one (1-1)

A function is one-to-one if different inputs map to different outputs. In mathematical terms, if \( f(x_1) = f(x_2) \) implies that \( x_1 = x_2 \).

1. Let's assume \( f(x_1) = f(x_2) \).
- This means \( |x_1| = |x_2| \).
2. The absolute value function gives the same output for both positive and negative inputs. For example:
- If \( x_1 = 1 \), then \( f(1) = |1| = 1 \).
- If \( x_2 = -1 \), then \( f(-1) = |-1| = 1 \).
3. Here, \( f(1) = f(-1) \) but \( 1 \neq -1 \).
4. Since we found two different inputs that give the same output, the function is **not one-to-one**.

### Step 2: Check if the function is onto (onto)

A function is onto if every element in the codomain (in this case, the set of real numbers \( \mathbb{R} \)) is the output of the function for some input from the domain.

1. The function \( f(x) = |x| \) produces only non-negative outputs.
2. Therefore, the range of \( f(x) \) is \( [0, \infty) \).
3. However, the codomain is \( \mathbb{R} \), which includes negative numbers.
4. Since there are no inputs \( x \) such that \( f(x) \) can yield a negative output, the function is **not onto**.

### Conclusion

Since the function \( f(x) = |x| \) is neither one-to-one nor onto, the correct answer is that the function is neither one-to-one nor onto.

### Final Answer

The correct option is: **The function is neither one-to-one nor onto.**

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