To analyze the problem, we are comparing two functions:

1. $f(x) = \frac{5}{2}x - 3$
2. $g(x) = -(x - 2)^2 + 4$

Step 1: Completing the Table

We evaluate $f(x)$ and $g(x)$ for the given values of $x$. I'll double-check and fill in the missing values:

For $f(x) = \frac{5}{2}x - 3$:

• $x = 2: f(2) = \frac{5}{2}(2) - 3 = 5 - 3 = 2$
• $x = 3: f(3) = \frac{5}{2}(3) - 3 = \frac{15}{2} - 3 = \frac{9}{2}$
• $x = 4: f(4) = \frac{5}{2}(4) - 3 = 10 - 3 = 7$

For $g(x) = -(x - 2)^2 + 4$:

• $x = 2: g(2) = -(2 - 2)^2 + 4 = 0 + 4 = 4$
• $x = 3: g(3) = -(3 - 2)^2 + 4 = -1 + 4 = 3$
• $x = 4: g(4) = -(4 - 2)^2 + 4 = -4 + 4 = 0$

Updated Table:

Step 2: Determining Where $f(x) = g(x)$

We need to find $x$ such that $f(x) = g(x)$. From the table:

• At $x = 2$: $f(2) = 2$ and $g(2) = 4$ → Not equal.
• At $x = 3$: $f(3) = \frac{9}{2} = 4.5$ and $g(3) = 3$ → Not equal.

Check between values:

1. Between $x = 2$ and $x = 3$, $f(x)$ increases from $2$ to $4.5$ and $g(x)$ decreases from $4$ to $3$. There is a crossing point where $f(x) = g(x)$. Solution exists between $x = 2$ and $x = 3$.
2. Between $x = 3$ and $x = 4$, $f(x)$ increases from $4.5$ to $7$ and $g(x)$ decreases from $3$ to $0$. No crossing.

Final Answer:

The solution to $f(x) = g(x)$ is between $x = 2$ and $x = 3$.

Would you like me to elaborate on solving these equations algebraically or clarify any step?

Follow-Up Questions:

1. How do we solve $\frac{5}{2}x - 3 = -(x - 2)^2 + 4$ algebraically?
2. What do the graphs of $f(x)$ and $g(x)$ look like, and where do they intersect?
3. How does the rate of change (slope) of $f(x)$ compare to $g(x)$ near their intersection?
4. Can we estimate the solution numerically using interpolation?
5. What happens to $f(x)$ and $g(x)$ as $x \to \infty$?

Tip: When comparing functions numerically, always test for trends (increasing vs. decreasing) to identify crossing intervals.