To determine whether an equation defines $y$ as a function of $x$, we need to check if, for each value of $x$, there is only one corresponding value of $y$. In other words, $y$ must pass the vertical line test, meaning no vertical line should intersect the graph of the equation more than once at any $x$-value. Let's examine each case:

(a) $x^2 + y = 25$

1. Solve for $y$:

   $y = 25 - x^2$

   This is a clear equation for $y$, and for any value of $x$, there is only one corresponding value of $y$. Thus, this equation defines $y$ as a function of $x$.

(b) $x + y^2 = 25$

1. Solve for $y$:

   $y^2 = 25 - x$ $y = \pm \sqrt{25 - x}$

   This shows that for each value of $x$, there are two possible values of $y$ (positive and negative square roots). Therefore, the equation does not define $y$ as a function of $x$ because a vertical line would intersect the graph at two points for certain $x$-values.

(c) $x = y^3$

1. Solve for $y$:

   $y = \sqrt[3]{x}$

   This equation gives a unique value of $y$ for each value of $x$. Since the cube root function is defined for all real numbers and is one-to-one, this equation defines $y$ as a function of $x$.

Summary:

• (a) $x^2 + y = 25$ → Yes, function.
• (b) $x + y^2 = 25$ → No, not a function.
• (c) $x = y^3$ → Yes, function.

Would you like further details on any of these?
Here are 5 related questions to explore:

1. How do you apply the vertical line test on a graph?
2. Why does solving for $y$ give multiple values in certain cases?
3. Can we use the horizontal line test to analyze one-to-one functions?
4. How do you determine if an equation involving absolute values defines a function?
5. What is the difference between one-to-one and onto functions?

Tip: A key to identifying functions is remembering that each input (x) should correspond to only one output (y).