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. $f(g(x))$: = 9x^2 + 12x + 4 + 5 = 9x^2 + 12x + 9$$
2. $g(f(x))$: = 3x^2 + 15 + 2 = 3x^2 + 17$$
3. $f(f(x))$: = x^4 + 10x^2 + 25 + 5 = x^4 + 10x^2 + 30$$

Column B:

1. $f(g(x))$: = 9x^2 + 12x + 4 + 5 = 9x^2 + 12x + 9$$
2. $g(f(x))$: = 3x^2 + 15 + 2 = 3x^2 + 17$$
3. $f(f(x))$: = 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.

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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.