The image contains two functions $f(x) = 4x - 3$ and $g(x) = \frac{1}{x}$, and you're tasked with finding the compositions $(f \circ g)(x)$ and $(g \circ f)(x)$, as well as determining the domain of each composite function.

Let's break down each part:

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

This is the composition $f(g(x))$, meaning we substitute $g(x)$ into $f(x)$.

Since $f(x) = 4x - 3$ and $g(x) = \frac{1}{x}$, we get: $f(g(x)) = f\left( \frac{1}{x} \right) = 4\left( \frac{1}{x} \right) - 3 = \frac{4}{x} - 3$

Domain of $f \circ g$:

For $f(g(x))$, $g(x) = \frac{1}{x}$ is defined for all $x \neq 0$, since division by zero is undefined. Therefore, the domain of $f(g(x))$ is: $x \in (-\infty, 0) \cup (0, \infty)$

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

This is the composition $g(f(x))$, meaning we substitute $f(x)$ into $g(x)$.

Since $g(x) = \frac{1}{x}$ and $f(x) = 4x - 3$, we get: $g(f(x)) = g(4x - 3) = \frac{1}{4x - 3}$

Domain of $g \circ f$:

For $g(f(x))$, the expression $\frac{1}{4x - 3}$ is undefined when $4x - 3 = 0$, which simplifies to $x = \frac{3}{4}$. Therefore, the domain of $g(f(x))$ is: $x \in (-\infty, \frac{3}{4}) \cup (\frac{3}{4}, \infty)$

Final Answer:

• $(f \circ g)(x) = \frac{4}{x} - 3$, with domain $x \in (-\infty, 0) \cup (0, \infty)$
• $(g \circ f)(x) = \frac{1}{4x - 3}$, with domain $x \in (-\infty, \frac{3}{4}) \cup (\frac{3}{4}, \infty)$

Would you like more details or have any questions on this?

Here are five related questions to deepen your understanding:

1. How would the composition change if $f(x)$ were a quadratic function?
2. What happens to the domain if we combine more than two functions?
3. Can you identify other functions that would restrict the domain even further?
4. How do you find the range of a composite function like $(f \circ g)(x)$?
5. What is the significance of the domain when graphing these composite functions?

Tip: Always check the domain of both the inner and outer functions when composing two functions. The result must satisfy the conditions of both.