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.

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!