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Funktionen är inte inverterbar om definitionsmängden inte begränsas.
vilka av följande begränsningar av definitionsmängden gör att funktionen blir inverterbar ?

The question is asking which restrictions on the domain make the function invertible. A function is invertible if it is **one-to-one** (i.e., each \( y \)-value is mapped from a unique \( x \)-value), meaning it must pass the **horizontal line test**.

From the graph provided, the function is not one-to-one over its entire domain because it fails the horizontal line test; there are multiple \( x \)-values for some \( y \)-values. However, by restricting the domain to specific intervals, we can make it one-to-one.

### Evaluating Each Interval
1. **\( -3 \leq x \leq 9 \):** This interval does not make the function one-to-one, as there are sections within this interval where a horizontal line intersects the curve more than once.

2. **\( x \geq 9 \):** This interval isolates a part of the curve where the function appears to be strictly decreasing, making it one-to-one.

3. **\( x < 4 \):** This interval includes parts of the curve where the function is not one-to-one, so it does not work.

4. **\( 4 \leq x \leq 9 \):** This interval appears to make the function one-to-one since the function is either strictly increasing or decreasing within this interval.

5. **\( x \geq 4 \):** This interval does not work, as the function is not one-to-one when \( x \geq 4 \) due to multiple points sharing the same \( y \)-values.

6. **\( x \leq -3 \):** This interval restricts the domain to a section of the curve that is strictly increasing, making the function one-to-one.

7. **\( -3 \leq x \leq 4 \):** This interval fails the horizontal line test due to multiple \( x \)-values mapping to the same \( y \)-values.

### Conclusion
The selections that make the function invertible are:
- \( x \geq 9 \)
- \( 4 \leq x \leq 9 \)
- \( x \leq -3 \)

These choices ensure that the function behaves in a strictly monotonic (increasing or decreasing) manner, allowing it to be inverted.

Would you like more details or explanations? 

Here are some follow-up questions for further exploration:
1. What does it mean for a function to pass the horizontal line test?
2. How do we determine if a function is one-to-one over a specific interval?
3. Can a function be invertible on different intervals even if it's not invertible over its entire domain?
4. What are the implications of restricting a function’s domain in real-world applications?
5. How does the concept of function inverses apply in calculus and algebra?

**Tip:** When checking if a function is invertible on an interval, look for sections where the function is strictly increasing or strictly decreasing.

Funktionen är inte inverterbar om definitionsmängden inte begränsas.
Vilka av följande begränsningar av definitionsmängden gör att funktionen blir inverterbar?

Understand which domain restrictions make a function invertible by ensuring it passes the horizontal line test. Learn about one-to-one functions, invertibility, and domain limitations in this detailed problem analysis.

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