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

1. Diagram Functions

Given:

• $f: A \to B$
• $g: B \to C$

a. Compute $(g \circ f)(1), (g \circ f)(2), (g \circ f)(3),$ and $(g \circ f)(4)$.

To solve this, follow these steps:

• First, evaluate $f(x)$ for each value from set $A$ to set $B$.
• Then, use the output from $f(x)$ as the input to $g(x)$.

From the diagram:

• $f(1) = a$, $f(2) = c$, $f(3) = b$, $f(4) = e$
• $g(a) = p$, $g(b) = r$, $g(c) = q$, $g(e) = q$

Thus:

• $(g \circ f)(1) = g(f(1)) = g(a) = p$
• $(g \circ f)(2) = g(f(2)) = g(c) = q$
• $(g \circ f)(3) = g(f(3)) = g(b) = r$
• $(g \circ f)(4) = g(f(4)) = g(e) = q$

b. Determine the domain and range of $g \circ f$.

• Domain: The domain of $g \circ f$ is the same as the domain of $f$, which is $A = \{1, 2, 3, 4\}$.
• Range: The range of $g \circ f$ is the set of possible outputs from $g \circ f$, which are $\{p, q, r\}$.

2. Composition of Functions

Now, let's move to the second part.

You are given four function pairs, and the task is to find $f \circ g$, $g \circ f$, and $f \cdot g$ (likely the multiplication of functions). Let's look at the first one as an example:

a.

• $f(x) = x^2 + 3$
• $g(x) = x - 1$

1. $f \circ g(x) = f(g(x)) = f(x - 1) = (x - 1)^2 + 3 = x^2 - 2x + 1 + 3 = x^2 - 2x + 4$
2. $g \circ f(x) = g(f(x)) = g(x^2 + 3) = (x^2 + 3) - 1 = x^2 + 2$
3. $f \cdot g(x) = f(x) \cdot g(x) = (x^2 + 3) \cdot (x - 1) = x^3 - x^2 + 3x - 3$

3. Finding Functions for Compositions

You are tasked with determining two functions $f$ and $g$ such that $f \circ g(x) = h(x)$.

a. Given $h(x) = (2x + 1)^2$, you can think of decomposing it as follows:

Let:

• $g(x) = 2x + 1$
• $f(x) = x^2$

Now check:

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

This means the functions $f(x) = x^2$ and $g(x) = 2x + 1$ satisfy $f \circ g(x) = h(x)$.

Do you want more details on any specific question or further steps? Here are 5 questions to deepen your understanding:

1. How do you determine the domain and range of a composed function?
2. What is the difference between function composition and function multiplication?
3. How do you decompose complex functions like in problem 3?
4. How can the composition of two functions simplify expressions in calculus?
5. What are the steps to find $f \circ g$ for functions with square roots or fractions?

Tip: Always check the domain restrictions when composing functions, especially when dealing with square roots or fractions.