To determine how many roots the functions $f(x) = x^2 - 9$ and $y = g(x)$ have in common, we need to find the points where the graphs of these functions intersect.

First, let's identify the roots of $f(x) = x^2 - 9$: $f(x) = x^2 - 9 = 0 \implies x^2 = 9 \implies x = \pm 3$ So, the roots of $f(x)$ are $x = 3$ and $x = -3$.

Now, looking at the graph of $y = g(x)$, we need to determine if these roots also lie on the graph of $g(x)$. We check the points $x = 3$ and $x = -3$ on the graph:

1. For $x = -3$, $y = g(-3) = 0$, which matches the root of $f(x)$.
2. For $x = 3$, $y = g(3) = 0$, which also matches the root of $f(x)$.

Thus, the two functions have two roots in common: $x = 3$ and $x = -3$.

Would you like any additional details or have any questions?

Here are 5 questions you might want to ask:

1. How do I determine the intersection points of two graphs analytically?
2. Can you explain how to solve quadratic equations in general?
3. What are some properties of quadratic functions?
4. How do I graph quadratic functions by hand?
5. What are the different forms of a quadratic equation and how do they relate?

Tip: When finding the roots of a function graphically, look for the points where the graph intersects the x-axis.