For f left parenthesis x right parenthesis equals 5 x plus 7 and g left parenthesis x right parenthesis equals x squared comma find the following composite functions and state the domain of each.
​(a)  fcircleg         ​ (b)  gcirclef           ​ (c)  fcirclef            ​ (d)  gcircleg

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.

For f(x) = 5x + 7 and g(x) = x^2, find the following composite functions and state the domain of each. (a) f ∘ g (b) g ∘ f (c) f ∘ f (d) g ∘ g

Explore how to find composite functions f ∘ g, g ∘ f, f ∘ f, and g ∘ g with the functions f(x) = 5x + 7 and g(x) = x^2. This problem includes a detailed step-by-step solution with an explanation of each function's domain. Suitable for Grades 10-12.

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