It looks like you've uploaded a photo containing a set of mathematical problems involving functions and compositions of functions. Let's go through some of the key problems and solutions.

Problem 16:

Given:

• $f: \mathbb{R} \to \mathbb{R}$
• $g: \mathbb{R} \to \mathbb{R}$
• $g(x) = 3x + 1$
• $(f \circ g)(x) = x^2 + 3$

Find $f(2x - 4)$.

To solve, we first need to determine $f(x)$ using the equation for $(f \circ g)(x)$.

$(f \circ g)(x) = f(g(x)) = f(3x + 1) = x^2 + 3$

Thus, $f(y) = \left(\frac{y - 1}{3}\right)^2 + 3$, where $y = 3x + 1$.

Substitute $y = 2x - 5$ into this formula, and simplify to get $f(2x - 4)$.

Problem 17:

We are given:

• $f(x) = 3x^2 - 4x + 6$
• $g(x) = 2x - 1$
• $(f \circ g)(x) = 101$

We are tasked with finding $x$.

Using the composition $(f \circ g)(x) = f(g(x))$, we substitute $g(x) = 2x - 1$ into $f(x)$:

$f(2x - 1) = 101$

Solve for $x$.

Problem 18:

Given:

• $f(x) = x^2 - x$
• $g(x) = 2x + 3$
• $h(x) = 1 - 3x$

Find $(f \circ g \circ h)(x)$, which involves evaluating $f(g(h(x)))$.

First, calculate $h(x)$, substitute into $g(x)$, then substitute the result into $f(x)$.

Would you like more detailed solutions to any specific problems, or should I continue solving others from the image?

Here are five follow-up questions based on the problems:

1. How do you find the composition of two functions, $f(g(x))$?
2. What is the step-by-step process for solving $(f \circ g)(x) = 101$?
3. How do you simplify complex expressions involving function compositions like $f(g(h(x)))$?
4. How can you derive $f(x)$ from $(f \circ g)(x)$?
5. What techniques are helpful when solving quadratic equations in function problems?

Tip: Always break down compositions of functions step by step to avoid errors, starting from the innermost function and working outward.