https://file.aiquickdraw.com/mathbot/file/17266803470233vuksb9v.jpg

Let's analyze each function to determine if it's one-to-one:

1. **(i) \( f(x) = (x + 6)^3 - 4 \)**:
   - The cubic function \( g(x) = (x+6)^3 \) is one-to-one because it passes the horizontal line test (it's strictly increasing or decreasing over its domain).
   - Shifting the function down by 4 units (the -4) does not affect the one-to-one property.
   - **Conclusion**: This function is **one-to-one**.

2. **(ii) \( f(x) = \frac{1}{7x} \)**:
   - This function is a rational function. The function \( f(x) = \frac{1}{x} \) is known to be one-to-one, as each value of \( x \) produces a unique output.
   - Multiplying by 1/7 does not affect the one-to-one property.
   - **Conclusion**: This function is **one-to-one**.

3. **(iii) \( f(x) = \begin{cases} 
x^2 + 3 & x \geq 0 \\
5x + 2 & x < 0 
\end{cases} \)**:
   - For \( x \geq 0 \), the function is \( f(x) = x^2 + 3 \), which is **not** one-to-one because the quadratic function does not pass the horizontal line test.
   - For \( x < 0 \), the function is linear, and linear functions are one-to-one.
   - However, due to the quadratic part \( x^2 + 3 \) for \( x \geq 0 \), the overall function is **not one-to-one**.
   
### Answer: \( \boxed{G} \) (i and ii only).

Would you like a detailed explanation of any of the steps? Here are 5 follow-up questions to consider:
1. Why does the horizontal line test determine whether a function is one-to-one?
2. What are some other transformations of functions that preserve the one-to-one property?
3. How can you visually determine if a quadratic function is one-to-one?
4. Why are piecewise functions often tricky when analyzing one-to-one properties?
5. Can a function be one-to-one on part of its domain but not overall?

**Tip**: Always apply the horizontal line test to check for one-to-one functions.

Which of the following functions are one-to-one?
(i) f(x) = (x + 6)^3 - 4
(ii) f(x) = 1/(7x)
(iii) f(x) = {x^2 + 3, x ≥ 0; 5x + 2, x < 0}
(A) none of them (B) all of them (C) (ii) only (D) (i) only (E) (ii) and (iii) only (F) (iii) only (G) (i) and (ii) only (H) (i) and (iii) only

Explore how to determine if functions are one-to-one by analyzing cubic, rational, and piecewise functions. This problem involves key algebraic concepts like the horizontal line test, one-to-one properties, and transformations. Ideal for high school students learning about functions.

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