https://file.aiquickdraw.com/mathbot/file/17273561048381q4drbaf.jpg

Let's break down the problem from the image.

The functions provided are:
- \( f(x) = x + 2 \)
- \( g(x) = x - 1 \)
- \( h(x) = x^2 \)
- \( k(x) = \frac{1}{2}x^2 \)

Given that \( x = 2 \), the task involves evaluating the functions at \( x = 2 \), and then performing a series of operations.

You have already found the following:

- \( f(2) = 2 + 2 = 4 \)
- \( g(2) = 2 - 1 = 1 \)
- \( h(2) = 2^2 = 4 \)
- \( k(2) = \frac{1}{2}(2^2) = \frac{1}{2}(4) = 2 \)

Now let's address the operations one by one:

1. **\( f(x) + g(x) \)**:
   - \( f(2) + g(2) = 4 + 1 = 5 \)

2. **\( g(x) - f(x) \)**:
   - \( g(2) - f(2) = 1 - 4 = -3 \)

3. **\( f(x) \cdot g(x) \)**:
   - \( f(2) \cdot g(2) = 4 \cdot 1 = 4 \)

4. **\( h(x) \cdot k(x) \)**:
   - \( h(2) \cdot k(2) = 4 \cdot 2 = 8 \)

5. **\( h(x) - k(x) \)**:
   - \( h(2) - k(2) = 4 - 2 = 2 \)

6. **\( \left( f(x) - g(x) \right)^2 \div \left( h(x) \cdot k(x) \right) \)**:
   - \( \left( f(2) - g(2) \right)^2 = (4 - 1)^2 = 3^2 = 9 \)
   - \( \left( h(2) \cdot k(2) \right) = 4 \cdot 2 = 8 \)
   - So, \( \frac{9}{8} = 1.125 \)

Thus, the complete results are:
1. \( f(2) + g(2) = 5 \)
2. \( g(2) - f(2) = -3 \)
3. \( f(2) \cdot g(2) = 4 \)
4. \( h(2) \cdot k(2) = 8 \)
5. \( h(2) - k(2) = 2 \)
6. \( \frac{(f(2) - g(2))^2}{h(2) \cdot k(2)} = 1.125 \)

Let me know if you'd like more details on any of these steps.

### Relative Questions:
1. What happens if \( x = 3 \)? How would the results change?
2. How would the answers change if one of the functions was modified, such as \( f(x) = x^3 \)?
3. Can you explain the importance of the order of operations in solving these?
4. How would these operations differ if we used different functions like trigonometric or logarithmic functions?
5. Can you think of a real-world application for such function operations?

### Tip:
Always double-check your work by plugging values back into the original functions to ensure the calculations are consistent.

Given the functions f(x) = x + 2, g(x) = x - 1, h(x) = x^2, and k(x) = (1/2)x^2, perform the indicated operations. The value of x is given as 2.

Learn how to perform operations on the given functions f(x), g(x), h(x), and k(x) when x = 2. Detailed steps for addition, subtraction, multiplication, and division of these functions.

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