The problem in the image involves two functions, $f(x) = -x^2 + 4x - k$ and $g(x) = x^2 - kx + 4$, and we are tasked with finding how many values of $k \in \mathbb{Z}$ allow these functions to intersect at most at one point.

Steps to Solve:

1. Set the functions equal to each other:
   To find the points of intersection, we need to solve the equation $f(x) = g(x)$. That gives:
   $-x^2 + 4x - k = x^2 - kx + 4$

2. Simplify the equation:
   Rearranging all terms to one side: $-x^2 + 4x - k - x^2 + kx - 4 = 0$ Combine like terms: $-2x^2 + (4 + k)x - (k + 4) = 0$

   This is a quadratic equation in terms of $x$, which we can solve using the quadratic formula.

3. Discriminant Condition:
   Since we are asked to find when the functions intersect at most at one point, we need the discriminant of this quadratic to be non-positive (i.e., $\Delta \leq 0$).

   The discriminant for a quadratic equation $ax^2 + bx + c = 0$ is given by: $\Delta = b^2 - 4ac$ Here, $a = -2$, $b = 4 + k$, and $c = -(k + 4)$.

   Substituting into the discriminant formula: $\Delta = (4 + k)^2 - 4(-2)(-(k + 4))$ Simplifying: $\Delta = (4 + k)^2 - 8(k + 4)$ Expand both terms: $\Delta = 16 + 8k + k^2 - 8k - 32$ Simplify: $\Delta = k^2 - 16$

4. Solve for the Discriminant:
   For at most one point of intersection, we need $\Delta \leq 0$. So: $k^2 - 16 \leq 0$ This simplifies to: $k^2 \leq 16$ Hence, $-4 \leq k \leq 4$.

5. Conclusion:
   The integer values of $k$ are $-4, -3, -2, -1, 0, 1, 2, 3, 4$. So, there are 9 possible values of $k$.

The final answer is 9.

Would you like further explanation on any step or need more details?

Here are 5 related questions for further practice:

1. What would happen if the discriminant was strictly greater than 0?
2. How does the number of intersections change if $\Delta > 0$?
3. Could these functions intersect at exactly two points? Why or why not?
4. What is the geometric interpretation of the condition $\Delta \leq 0$?
5. How would the solution change if one of the functions had an additional linear term?

Tip: When solving for intersections of curves, always check the discriminant of the resulting quadratic equation—this gives you the number of intersection points.