determine the a part if it is a function or not

Let's evaluate each part of Section A to determine whether each expression represents a function. 

### Section A: Functions or Not?

1. **\( f(x) = x^2 \)**
   - **Function:** This is a quadratic function. For every input \( x \), there is a unique output \( f(x) \).

2. **\( g(x) = \sqrt{x} \)**
   - **Function:** This is a square root function. For every non-negative input \( x \), there is a unique output \( g(x) \). However, it is only defined for \( x \geq 0 \).

3. **\( f(x) = x^3 \)**
   - **Function:** This is a cubic function. For every input \( x \), there is a unique output \( f(x) \).

4. **\( f(x) = |2x + 9| \)**
   - **Function:** This is an absolute value function. For every input \( x \), there is a unique output \( f(x) \).

5. **\( x = y^2 \)**
   - **Not a Function:** This equation represents a sideways parabola. For a given \( x \), there can be two possible values of \( y \) (one positive and one negative), which means it does not pass the vertical line test. 

6. **\( y = \frac{3x}{x - 3} \)**
   - **Function:** This is a rational function. For every input \( x \neq 3 \), there is a unique output \( y \). It is not defined at \( x = 3 \), but otherwise, it behaves like a function.

7. **\( y = \frac{3}{x + 3} \)**
   - **Function:** This is another rational function. For every input \( x \neq -3 \), there is a unique output \( y \). It is not defined at \( x = -3 \), but otherwise, it behaves like a function.

8. **\( f(x) = -x \)**
   - **Function:** This is a linear function. For every input \( x \), there is a unique output \( f(x) \).

### Summary:

- Functions: \( f(x) = x^2 \), \( g(x) = \sqrt{x} \), \( f(x) = x^3 \), \( f(x) = |2x + 9| \), \( y = \frac{3x}{x - 3} \), \( y = \frac{3}{x + 3} \), \( f(x) = -x \).
- Not a Function: \( x = y^2 \).

Would you like further details on any of these conclusions?

### Related Questions
1. How do you apply the vertical line test to determine if a graph represents a function?
2. What characteristics make a function different from a mere relation?
3. Can a rational expression represent a function?
4. How does the equation \( x = y^2 \) differ from \( y = x^2 \) in terms of being a function?
5. What happens to a function when the denominator in a rational function equals zero?

### Tip
In mathematics, a function assigns exactly one output for each input. If any input maps to more than one output, it is not a function.

Explore the concept of functions and how to determine if an equation represents a function using the vertical line test. This page discusses various examples including quadratic, cubic, absolute value, and rational functions. Ideal for students studying functions in Grades 9-12.

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