Let's break down the problem and analyze the given information step by step.

Given:

• The function $f(x) = x^2 - 4x - 5$ is a quadratic function.
• We have a graph of another function $g(x)$, which is a parabola that opens upwards, with the following characteristics:
  • The y-intercept is at $(0, 9)$, meaning $g(0) = 9$.
  • The x-intercepts are at $x = 2$ and $x = 5$, meaning $g(2) = 0$ and $g(5) = 0$.
  • The minimum is at approximately $(3.5, -2)$, suggesting the vertex of the parabola is near $x = 3.5$ and $g(x)$ reaches its lowest value there.

Step 1: Find the roots of $f(x)$

We start by factoring $f(x) = x^2 - 4x - 5$:

$f(x) = (x - 5)(x + 1)$

So, the roots of $f(x)$ are: $x = 5 \quad \text{and} \quad x = -1$

Step 2: Analyze the roots of $g(x)$

From the graph description of $g(x)$, we know the x-intercepts are at $x = 2$ and $x = 5$. This means the roots of $g(x)$ are: $x = 2 \quad \text{and} \quad x = 5$

Step 3: Compare the roots of $f(x)$ and $g(x)$

• $f(x)$ has roots at $x = 5$ and $x = -1$.
• $g(x)$ has roots at $x = 2$ and $x = 5$.

The only root that $f(x)$ and $g(x)$ have in common is $x = 5$.

Conclusion:

The functions $f(x)$ and $g(x)$ share one root in common, and each has another root that is not shared.

Thus, the correct answer is: (Choice B) $f$ and $g$ share one root in common but each have another root that is not shared.

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Would you like a further explanation or help with another question? Here's a list of related topics to explore:

1. How to factor quadratic equations.
2. Understanding the vertex form of a parabola.
3. Solving for the roots of quadratic functions.
4. The connection between the graph of a quadratic and its factors.
5. How to interpret and sketch quadratic graphs.

Tip: When factoring quadratics, always check if the coefficients are easily factorable, and if not, try the quadratic formula!