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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.

Suppose the function f(x) = -x^2 + 4x - k and g(x) = x^2 - kx + 4 intersects at most at one point, where k ∈ Z. Find the possible number of values of k.

Solve a problem involving two quadratic functions f(x) = -x^2 + 4x - k and g(x) = x^2 - kx + 4. Learn how to determine the number of values of k where these functions intersect at most at one point. This involves key concepts from algebra and quadratic equations.

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