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The question in the image asks:

"If \( y = f(x) = \frac{x^2}{1 + x^2} \), is the function one-one and onto, provided \( f: \mathbb{R} \to \mathbb{R} \)?"

Let's analyze the function \( f(x) = \frac{x^2}{1 + x^2} \) in terms of whether it is **one-to-one (injective)** and **onto (surjective)**.

### Step 1: Checking if \( f(x) \) is One-to-One

A function is one-to-one if each value in the range corresponds to exactly one value in the domain. We can check this by determining if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \).

Let's rewrite \( f(x) \):
\[
f(x) = \frac{x^2}{1 + x^2}
\]

Notice that \( f(x) \) depends on \( x^2 \), meaning it is **symmetric** around the y-axis. Thus, \( f(x) = f(-x) \), indicating that the function is not one-to-one because \( f(x_1) = f(x_2) \) does not imply \( x_1 = x_2 \); \( x \) and \( -x \) yield the same value. For example, \( f(1) = f(-1) \).

Therefore, **\( f(x) \) is not one-to-one**.

### Step 2: Checking if \( f(x) \) is Onto

To check if \( f(x) \) is onto (surjective) with \( f: \mathbb{R} \to \mathbb{R} \), we need to determine if every real number \( y \in \mathbb{R} \) has a corresponding \( x \in \mathbb{R} \) such that \( f(x) = y \).

Since \( f(x) = \frac{x^2}{1 + x^2} \), observe the following:
1. \( x^2 \geq 0 \), so \( f(x) \geq 0 \) for all \( x \).
2. As \( x \to \pm \infty \), \( f(x) \to 1 \).
3. As \( x \to 0 \), \( f(x) \to 0 \).

This shows that the range of \( f(x) \) is \( [0, 1) \), not all of \( \mathbb{R} \). Thus, **\( f(x) \) is not onto** if we consider the codomain as \( \mathbb{R} \).

### Conclusion

The function \( f(x) = \frac{x^2}{1 + x^2} \), with \( f: \mathbb{R} \to \mathbb{R} \), is **neither one-to-one nor onto**.

Would you like a more detailed explanation, or do you have any other questions?

### Additional Questions for Practice:
1. What would be the range of \( f(x) = \frac{x^2}{1 + x^2} \) if the codomain is restricted to \( [0,1) \)?
2. Can you find the derivative of \( f(x) = \frac{x^2}{1 + x^2} \) to determine if it's increasing or decreasing?
3. How does the symmetry of a function affect whether it is one-to-one?
4. What changes could make \( f(x) \) onto \( [0,1] \)?
5. Can you think of a way to restrict the domain of \( f(x) \) to make it one-to-one?

### Tip:
To test if a function is one-to-one, use the Horizontal Line Test: if any horizontal line intersects the graph more than once, the function is not one-to-one.

If y = f(x) = \frac{x^2}{1 + x^2}, is the function one-one and onto, provided f: \mathbb{R} \to \mathbb{R}?

Explore whether the function f(x) = x^2/(1 + x^2) is one-to-one and onto over the domain and codomain of all real numbers. This problem delves into function properties, symmetry, and injective/surjective criteria, making it ideal for advanced high school math students.

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