The text in the image is written in Hebrew and poses a mathematical question related to a function $f : \mathbb{R} \to \mathbb{R}$. The question appears to be asking whether there exists a surjective function (פונקציה רציפה) where each value is reached exactly twice (המקבלת כל ערך בדיוק פעמיים).

This question likely belongs to the context of real analysis or continuous functions. Let’s analyze the problem:

Analysis of the Problem:

1. Surjective Function: A function $f: \mathbb{R} \to \mathbb{R}$ is surjective if, for every $y \in \mathbb{R}$, there exists at least one $x \in \mathbb{R}$ such that $f(x) = y$.

2. Condition of "Exactly Twice": The function must map every $y \in \mathbb{R}$ to exactly two distinct $x$-values in the domain $\mathbb{R}$. That is, if $f(x_1) = y$ and $f(x_2) = y$, then $x_1 \neq x_2$, and no other $x$ maps to $y$.

3. Continuous Function: Continuity adds an additional restriction: $f(x)$ cannot have jumps or discontinuities.

Possible Answer:

Yes, it is possible to construct such a function. One example of such a function is a parabola that is inverted and opens downwards. A specific example would be:

$f(x) = a - x^2$

Why this works:

• This function is continuous everywhere because it is a polynomial.
• It is surjective if $a = +\infty$, as it spans all real numbers $\mathbb{R}$.
• For any given $y$, there are exactly two solutions to the equation $a - x^2 = y$, specifically: $x = \pm \sqrt{a - y}$

However, for this specific parabola to "fit" the problem, additional constraints (like its domain or behavior at infinities) may need to be addressed.

Would you like me to construct a more general proof, or would you like more specific examples?

5 Related Questions:

1. What is the formal definition of a surjective function?
2. How can we determine if a given function maps all $y$ values exactly twice?
3. Can we construct a non-polynomial function that satisfies these conditions?
4. What are the implications of the continuity constraint on such functions?
5. How does the function's symmetry help in satisfying the "exactly twice" condition?

One Tip:

When analyzing functions with specific properties, graphing examples can provide valuable intuition and confirm whether they meet the problem's requirements.