Given the functions $f(x) = x^3 + 1$ and $g(x) = \frac{1}{x}$, let's calculate the values for each of the specified operations:

(a) $(f + g)(x)$

$(f + g)(x) = f(x) + g(x)$

$= (x^3 + 1) + \frac{1}{x}$

$= x^3 + 1 + \frac{1}{x}$

(b) $(f - g)(2x)$

First, find $f(2x)$ and $g(2x)$:

$f(2x) = (2x)^3 + 1 = 8x^3 + 1$

$g(2x) = \frac{1}{2x}$

Now, calculate $(f - g)(2x)$:

$(f - g)(2x) = f(2x) - g(2x) = (8x^3 + 1) - \frac{1}{2x}$

$= 8x^3 + 1 - \frac{1}{2x}$

(c) $(f \circ g)(x)$

$(f \circ g)(x) = f(g(x))$:

$f(g(x)) = f\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^3 + 1$

$= \frac{1}{x^3} + 1$

(d) $(g \circ f)(x)$

$(g \circ f)(x) = g(f(x))$:

$g(f(x)) = g(x^3 + 1) = \frac{1}{x^3 + 1}$

(e) $((f - g) \circ g)(x)$

$((f - g) \circ g)(x) = (f - g)(g(x))$:

1. Find $f(g(x))$ and $g(g(x))$:

$f(g(x)) = f\left(\frac{1}{x}\right) = \frac{1}{x^3} + 1$

$g(g(x)) = g\left(\frac{1}{x}\right) = x$

2. Now calculate:

$(f - g)(g(x)) = f(g(x)) - g(g(x))$

$= \left(\frac{1}{x^3} + 1\right) - x$

$= \frac{1}{x^3} + 1 - x$

Would you like more details or have any questions?

Here are some related questions to expand on this topic:

1. How do you find the domain of a composite function?
2. What happens to the range of a function when it's composed with another function?
3. How can you determine if a composite function is invertible? 4