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Let's examine the given options to determine which of the following is a one-to-one function.

### Definitions:
- A **one-to-one function** (or **injective function**) is a function in which every element of the codomain is mapped to by at most one element of the domain. In simpler terms, no two different inputs (x-values) produce the same output (y-value).

Now, let's analyze each option:

#### Option 1: \( 2x = -3x^2 + y \)
- This is a quadratic equation (due to the \(x^2\) term). Quadratic functions are not one-to-one because they fail the horizontal line test—multiple values of \(x\) can result in the same value of \(y\). 
- **Not one-to-one.**

#### Option 2: \( x + 3 = y^2 - 7 \)
- Rearranging this gives \( y^2 = x + 10 \). 
- This is a quadratic equation in terms of \(y\), which means it is not one-to-one because multiple values of \(y\) can result from the same \(x\) value (for example, both \(y = 3\) and \(y = -3\) give the same result).
- **Not one-to-one.**

#### Option 3: \( -\frac{4}{5}x = 3y \)
- Solving for \(y\), we get \( y = -\frac{4}{15}x \). 
- This is a linear equation, and linear functions of this form are always one-to-one since no two values of \(x\) will produce the same \(y\) value.
- **This is a one-to-one function.**

#### Option 4: \( x^3 + 3x = |y - 8| + 4 \)
- The absolute value function (\(|y - 8|\)) is not one-to-one because an absolute value function gives the same result for both positive and negative values of its input (for instance, \(|3| = 3\) and \(|-3| = 3\)).
- **Not one-to-one.**

### Conclusion:
The correct answer is **Option 3**: \( -\frac{4}{5}x = 3y \), which represents a one-to-one function.

Do you have any questions or want more details?

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Here are some related questions:
1. How do you test whether a function is one-to-one using the horizontal line test?
2. Can a function be one-to-one but not onto?
3. What are examples of non-linear one-to-one functions?
4. What is the difference between one-to-one and onto functions?
5. How does the concept of an inverse function relate to one-to-one functions?

**Tip:** A one-to-one function must pass the horizontal line test, meaning no horizontal line should intersect the graph more than once.

Which of the following is one-to-one function? 
1. 2x = -3x^2 + y 
2. x + 3 = y^2 - 7 
3. -4/5x = 3y 
4. x^3 + 3x = |y - 8| + 4

Analyze which function is one-to-one from given options including linear, quadratic, and absolute value functions. Understand the application of the horizontal line test and key algebraic concepts for high school students.

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