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Let's go through each of the functions from Exercises 7 to 10 to determine if they are one-to-one. A function is one-to-one if it passes the **horizontal line test**, meaning no horizontal line intersects the graph of the function more than once.

### 7. \( f(x) = \begin{cases} 
3 - x, & x < 0 \\
3, & x \geq 0 
\end{cases} \)
- **For \(x < 0\)**: \(f(x) = 3 - x\) is a linear function with a slope of -1, which is one-to-one in this domain.
- **For \(x \geq 0\)**: \(f(x) = 3\) is a constant function, which is not one-to-one because all values of \(x \geq 0\) map to the same \(f(x) = 3\).

Thus, this function is **not one-to-one** because of the constant part in the second piece.

### 8. \( f(x) = \begin{cases} 
2x + 6, & x \leq -3 \\
x + 4, & x > -3 
\end{cases} \)
- **For \(x \leq -3\)**: \(f(x) = 2x + 6\) is linear with slope 2, so it's one-to-one.
- **For \(x > -3\)**: \(f(x) = x + 4\) is also linear with slope 1, making it one-to-one.

Since both pieces are linear and have different slopes, they cannot overlap and thus the function is **one-to-one**.

### 9. \( f(x) = \begin{cases} 
1 - \frac{x}{2}, & x \leq 0 \\
\frac{x}{x + 2}, & x > 0 
\end{cases} \)
- **For \(x \leq 0\)**: \(f(x) = 1 - \frac{x}{2}\) is linear with a slope of -1/2, which is one-to-one.
- **For \(x > 0\)**: \(f(x) = \frac{x}{x + 2}\) is a rational function. To check if this piece is one-to-one, observe its derivative:
  \[
  \frac{d}{dx} \left(\frac{x}{x+2}\right) = \frac{2}{(x+2)^2}
  \]
  Since the derivative is always positive for \(x > 0\), this portion is increasing and thus one-to-one.

Since both parts are one-to-one, the entire function is **one-to-one**.

### 10. \( f(x) = \begin{cases} 
2 - x^2, & x \leq 1 \\
x^2, & x > 1 
\end{cases} \)
- **For \(x \leq 1\)**: \(f(x) = 2 - x^2\) is a quadratic function that opens downward. It is not one-to-one because the same \(f(x)\) value can be obtained from two different \(x\) values (symmetry around the vertex).
- **For \(x > 1\)**: \(f(x) = x^2\) is a quadratic function that opens upward and is not one-to-one for the same reason.

Thus, this function is **not one-to-one**.

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Would you like further explanation on any of these? Here are some related questions for practice:

1. How do you apply the horizontal line test to check if a function is one-to-one?
2. What are some common functions that are always one-to-one?
3. How does a piecewise function affect the determination of whether a function is one-to-one?
4. How can the derivative be used to determine if a function is one-to-one on a given interval?
5. What characteristics of quadratic functions generally make them not one-to-one?

**Tip:** Always check if different pieces of a piecewise function overlap by evaluating their behavior at the boundaries!

In Exercises 7-10, determine from its graph if the function is one-to-one.

Learn how to determine if a function is one-to-one using the horizontal line test and derivative analysis. This problem involves piecewise functions, linear, quadratic, and rational functions, suitable for students in Grades 10-12.

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