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.