Let's work through the problems in the image step by step.

1. Problem 1

Given:

• $f = \{(-5,1), (-2,8), (-1,3), (4,10), (6,-4), (8,6)\}$
• $g = \{(-4,2), (1,0), (6,-1), (8,-5), (10,-2)\}$
• $h = \{(-5,12), (-4,1), (-3,5), (-1,0), (2,10), (4,8), (6,-4)\}$

We need to determine:

• a) $(f \circ g \circ h)(x)$
• b) $(h \circ g \circ f)(x)$

Let's start by evaluating these composite functions.

a) $(f \circ g \circ h)(x)$ means:

• First, evaluate $h(x)$
• Then, evaluate $g(h(x))$
• Finally, evaluate $f(g(h(x)))$

b) $(h \circ g \circ f)(x)$ means:

• First, evaluate $f(x)$
• Then, evaluate $g(f(x))$
• Finally, evaluate $h(g(f(x)))$

Now, you can plug values into these compositions based on the defined points.

2. Problem 2

Given:

• $f(x) = 2x^2 - 5x$
• $g(x) = \frac{10}{2x+1}$

We need to determine:

• a) $(f + g)(3x)$
• b) $\left( \frac{f}{g} \right)(2)$
• c) $(f \cdot g)(-1)$

Let’s calculate these step by step:

• a) $(f + g)(3x)$ means $f(3x) + g(3x)$
• b) $\frac{f(2)}{g(2)}$
• c) $f(-1) \cdot g(-1)$

3. Problem 3

Given:

• $f(x) = 3x + 5$
• $g(x) = 2x - 7$

We need to determine $(f \circ g)(x)$, which is $f(g(x))$.

4. Problem 4

Given:

• $f(x) = 2x + 2$
• $g(x) = \sqrt{4 - x}$
• $h(x) = 4x - x^2$

We need to determine $(g \circ f \circ h)(x)$.

This means:

• First, evaluate $h(x)$
• Then, evaluate $f(h(x))$
• Finally, evaluate $g(f(h(x)))$

5. Problem 5

Given:

• $f(x) = 2x + 5$
• $g(x) = x^2$
• $h(x) = \frac{-x}{x-4}$

We need to determine $(f \circ g \circ h)(6)$, which is:

• First, evaluate $h(6)$
• Then, evaluate $g(h(6))$
• Finally, evaluate $f(g(h(6)))$

Would you like to go over the detailed steps of solving each problem? If so, which specific problem would you like me to break down further?

5 Questions:

1. What does it mean to compose functions like $f \circ g \circ h$?
2. How do we approach evaluating composite functions from the inside out?
3. How does substitution work when functions are given as sets of points rather than equations?
4. What techniques help simplify expressions involving multiple composite functions?
5. How can we apply composite function operations to solve real-world problems?

Tip:

When dealing with composite functions, always start from the innermost function and work your way outward to ensure accuracy.