https://file.aiquickdraw.com/mathbot/file/172490068869354kshg89.jpg

Let's solve the problems from the image.

### Problem 14:
Given \( f(x) = \sqrt{x} - \frac{1}{2}x^2 + 2 \) and \( g(x) = 1 - \ln(x) \).

You are asked to find \( f(g(x)) \).

#### Solution:
To find \( f(g(x)) \), substitute \( g(x) \) into \( f(x) \):

\[
f(g(x)) = \sqrt{g(x)} - \frac{1}{2}(g(x))^2 + 2
\]

Substituting \( g(x) = 1 - \ln(x) \):

\[
f(g(x)) = \sqrt{1 - \ln(x)} - \frac{1}{2}(1 - \ln(x))^2 + 2
\]

So the final expression for \( f(g(x)) \) is:

\[
f(g(x)) = \sqrt{1 - \ln(x)} - \frac{1}{2}(1 - \ln(x))^2 + 2
\]

### Problem 15:
Given \( f(x) = x^2 + 5 \) and \( g(x) = 3x + 2 \).

You're asked to compare the calculations in Column A and Column B and determine if there is a mistake.

#### Column A:
1. **\( f(g(x)) \)**:
   \[
   f(g(x)) = (3x + 2)^2 + 5
   = 9x^2 + 12x + 4 + 5
   = 9x^2 + 12x + 9
   \]

2. **\( g(f(x)) \)**:
   \[
   g(f(x)) = 3(x^2 + 5) + 2
   = 3x^2 + 15 + 2
   = 3x^2 + 17
   \]

3. **\( f(f(x)) \)**:
   \[
   f(f(x)) = (x^2 + 5)^2 + 5
   = x^4 + 10x^2 + 25 + 5
   = x^4 + 10x^2 + 30
   \]

#### Column B:
1. **\( f(g(x)) \)**:
   \[
   f(g(x)) = (3x + 2)^2 + 5
   = 9x^2 + 12x + 4 + 5
   = 9x^2 + 12x + 9
   \]

2. **\( g(f(x)) \)**:
   \[
   g(f(x)) = 3(x^2 + 5) + 2
   = 3x^2 + 15 + 2
   = 3x^2 + 17
   \]

3. **\( f(f(x)) \)**:
   \[
   f(f(x)) = (x^2 + 5)^2 + 5
   = x^4 + 10x^2 + 25 + 5
   = x^4 + 10x^2 + 30
   \]

### Conclusion:
Both columns A and B have the same calculations. Therefore, there is **no mistake** in either column.

The correct answer is **D: Neither Column A nor B**.

---

Would you like further explanations or have any questions?

### Related Questions:
1. What is the significance of the composition of functions in calculus?
2. How would the results change in Problem 15 if \( g(x) = x^2 + 1 \) instead?
3. How can we verify that \( f(g(x)) \) is correctly computed?
4. What is the domain of the composed function \( f(g(x)) \) in Problem 14?
5. Can \( f(f(x)) \) be factored or simplified further in Problem 15?

### Tip:
When dealing with composition of functions, always ensure that the inner function's output is within the domain of the outer function.

Explore function composition problems with detailed examples and solutions. Includes solving complex compositions involving square root and logarithmic functions. Suitable for advanced high school students.

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