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.