Given the functions $f(x) = 5x + 7$ and $g(x) = x^2$, let's find the composite functions and their domains:

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

This represents $f(g(x))$, which means we substitute $g(x)$ into $f(x)$: $(f \circ g)(x) = f(g(x)) = f(x^2) = 5(x^2) + 7 = 5x^2 + 7.$

Domain of $(f \circ g)$:
Since $g(x) = x^2$ has a domain of all real numbers $\mathbb{R}$, and the expression $5x^2 + 7$ is defined for all real numbers, the domain of $(f \circ g)(x)$ is also all real numbers: $\text{Domain: } (-\infty, \infty).$

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

This represents $g(f(x))$, which means we substitute $f(x)$ into $g(x)$: $(g \circ f)(x) = g(f(x)) = g(5x + 7) = (5x + 7)^2.$

Domain of $(g \circ f)$:
Since $f(x) = 5x + 7$ has a domain of all real numbers $\mathbb{R}$, and the expression $(5x + 7)^2$ is defined for all real numbers, the domain of $(g \circ f)(x)$ is also all real numbers: $\text{Domain: } (-\infty, \infty).$

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

This represents $f(f(x))$, which means we substitute $f(x)$ into itself: $(f \circ f)(x) = f(f(x)) = f(5x + 7) = 5(5x + 7) + 7 = 25x + 35 + 7 = 25x + 42.$

Domain of $(f \circ f)$:
Since $f(x) = 5x + 7$ has a domain of all real numbers $\mathbb{R}$, and the expression $25x + 42$ is defined for all real numbers, the domain of $(f \circ f)(x)$ is also all real numbers: $\text{Domain: } (-\infty, \infty).$

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

This represents $g(g(x))$, which means we substitute $g(x)$ into itself: $(g \circ g)(x) = g(g(x)) = g(x^2) = (x^2)^2 = x^4.$

Domain of $(g \circ g)$:
Since $g(x) = x^2$ has a domain of all real numbers $\mathbb{R}$, and the expression $x^4$ is defined for all real numbers, the domain of $(g \circ g)(x)$ is also all real numbers: $\text{Domain: } (-\infty, \infty).$

Summary of Composite Functions and Their Domains

1. $(f \circ g)(x) = 5x^2 + 7$, Domain: $(-\infty, \infty)$
2. $(g \circ f)(x) = (5x + 7)^2$, Domain: $(-\infty, \infty)$
3. $(f \circ f)(x) = 25x + 42$, Domain: $(-\infty, \infty)$
4. $(g \circ g)(x) = x^4$, Domain: $(-\infty, \infty)$

Would you like a more detailed explanation of any of these?

Here are 5 related questions for further exploration:

1. What is the range of each composite function?
2. How do you determine the domain of a composite function in general?
3. What happens if one of the functions has a restricted domain?
4. Can you find the inverse of any of these composite functions?
5. How would the domains change if $g(x)$ were $\sqrt{x}$ instead of $x^2$?

Tip: When finding the domain of a composite function, always ensure that the input of the inner function is within its domain and that the resulting value is within the domain of the outer function.