We are tasked with solving the compositions of functions $f(x) = \frac{1}{x}$ and $g(x) = 2x + 1$, specifically:

1. $(f \circ g)(x)$
2. $(g \circ f)(x)$
3. $(f \circ f)(x)$
4. $(g \circ g)(x)$

Additionally, we will state the domains of each composition. Let’s solve them step by step:

1. $(f \circ g)(x) = f(g(x))$

Substitute $g(x) = 2x + 1$ into $f(x) = \frac{1}{x}$: $f(g(x)) = f(2x + 1) = \frac{1}{2x + 1}.$

Domain: For $f(g(x))$ to be defined, $2x + 1 \neq 0$.
Solve $2x + 1 = 0$: $x = -\frac{1}{2}$.
Thus, the domain is: $x \in \mathbb{R}, x \neq -\frac{1}{2}.$

2. $(g \circ f)(x) = g(f(x))$

Substitute $f(x) = \frac{1}{x}$ into $g(x) = 2x + 1$: $g(f(x)) = g\left(\frac{1}{x}\right) = 2\left(\frac{1}{x}\right) + 1 = \frac{2}{x} + 1.$

Domain: For $g(f(x))$ to be defined, $x \neq 0$ (since $f(x) = \frac{1}{x}$ requires $x \neq 0$).
Thus, the domain is: $x \in \mathbb{R}, x \neq 0.$

3. $(f \circ f)(x) = f(f(x))$

Substitute $f(x) = \frac{1}{x}$ into itself: $f(f(x)) = f\left(\frac{1}{x}\right) = \frac{1}{\frac{1}{x}} = x.$

Domain: For $f(f(x))$, $f(x) = \frac{1}{x}$ must be defined, so $x \neq 0$.
Thus, the domain is: $x \in \mathbb{R}, x \neq 0.$

4. $(g \circ g)(x) = g(g(x))$

Substitute $g(x) = 2x + 1$ into itself: $g(g(x)) = g(2x + 1) = 2(2x + 1) + 1 = 4x + 2 + 1 = 4x + 3.$

Domain: $g(g(x))$ is a linear function and is defined for all $x$.
Thus, the domain is: $x \in \mathbb{R}.$

Final Results:

1. $(f \circ g)(x) = \frac{1}{2x + 1}, \; \text{Domain: } x \in \mathbb{R}, x \neq -\frac{1}{2}$
2. $(g \circ f)(x) = \frac{2}{x} + 1, \; \text{Domain: } x \in \mathbb{R}, x \neq 0$
3. $(f \circ f)(x) = x, \; \text{Domain: } x \in \mathbb{R}, x \neq 0$
4. $(g \circ g)(x) = 4x + 3, \; \text{Domain: } x \in \mathbb{R}$

Would you like further details or explanations for any of these?

Related Questions:

1. What happens to the domains of $f \circ g(x)$ and $g \circ f(x)$ if $g(x) = x^2$ instead?
2. Can we interpret the domains graphically for these functions?
3. How does composition of functions change if one function is not one-to-one?
4. What are the inverse functions of $f(x) = \frac{1}{x}$ and $g(x) = 2x + 1$?
5. How do we find the composition of three functions $(f \circ g \circ h)(x)$?

Tip:

Always verify the domain of the inner function when solving composition problems, as restrictions propagate through the composition!