A function \( f(x) = 2 - 2x^2 \) is **not one-to-one**. 

### Explanation:
A function is **one-to-one** if each value in the range corresponds to exactly one unique value in the domain. In other words, if \( f(a) = f(b) \), then \( a = b \). 

#### Step-by-step check:
1. **Function:**
   \[ f(x) = 2 - 2x^2 \]
   
2. **Check if \( f(x) \) is one-to-one using algebra:**
   Assume \( f(a) = f(b) \) for two different inputs \( a \) and \( b \).

   \[ 2 - 2a^2 = 2 - 2b^2 \]

   Subtract 2 from both sides:
   \[ -2a^2 = -2b^2 \]

   Divide by -2:
   \[ a^2 = b^2 \]

   This equation implies that:
   \[ a = \pm b \]

   So, two different inputs (e.g., \( a = 2 \) and \( b = -2 \)) can produce the same output. Thus, the function is **not one-to-one**.

3. **Graphical confirmation:**
   The function \( f(x) = 2 - 2x^2 \) is a downward-facing parabola. Since it is symmetric about the y-axis, multiple \( x \)-values (one positive and one negative) can map to the same \( y \)-value. This symmetry confirms that the function is not one-to-one.

### Conclusion:
The function \( f(x) = 2 - 2x^2 \) is **not one-to-one** because different inputs can yield the same output.

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Would you like me to explore any other aspects of this function?

### 5 Related Questions:
1. How do you test if a function is **onto (surjective)**?
2. What is the **inverse function** of a one-to-one function?
3. Can the function \( f(x) = 2 - 2x^2 \) be restricted to a one-to-one part of its domain?
4. How does the **horizontal line test** help identify one-to-one functions?
5. What are the domain and range of \( f(x) = 2 - 2x^2 \)?

**Tip:** A quick way to test if a function is one-to-one is by applying the **horizontal line test** on its graph! If any horizontal line intersects the graph more than once, the function is not one-to-one.

Explore whether the function f(x) = 2 - 2x^2 is one-to-one. This algebra problem involves checking the one-to-one property using algebraic methods and the horizontal line test. Learn how different inputs can yield the same output in this quadratic function. Suitable for students in Grades 9-12.

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