If ​f(x)equals4 x plus 4 and ​g(x)equalsx minus 4​, find ​(fcircle​g)(x) and ​(gcircle​f)(x).
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Part 1
​(fcircle​g)(x)equals
  
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To solve for \( (f \circ g)(x) \) and \( (g \circ f)(x) \), we need to compute the composition of the functions \( f(x) = 4x + 4 \) and \( g(x) = x - 4 \).

### Part 1: \( (f \circ g)(x) \)
By definition:
\[
(f \circ g)(x) = f(g(x))
\]

1. Substitute \( g(x) = x - 4 \) into \( f(x) = 4x + 4 \):
   \[
   f(g(x)) = f(x - 4)
   \]

2. Replace \( x \) in \( f(x) = 4x + 4 \) with \( x - 4 \):
   \[
   f(x - 4) = 4(x - 4) + 4
   \]

3. Simplify:
   \[
   f(x - 4) = 4x - 16 + 4 = 4x - 12
   \]

Thus:
\[
(f \circ g)(x) = 4x - 12
\]

### Part 2: \( (g \circ f)(x) \)
By definition:
\[
(g \circ f)(x) = g(f(x))
\]

1. Substitute \( f(x) = 4x + 4 \) into \( g(x) = x - 4 \):
   \[
   g(f(x)) = g(4x + 4)
   \]

2. Replace \( x \) in \( g(x) = x - 4 \) with \( 4x + 4 \):
   \[
   g(4x + 4) = (4x + 4) - 4
   \]

3. Simplify:
   \[
   g(4x + 4) = 4x + 4 - 4 = 4x
   \]

Thus:
\[
(g \circ f)(x) = 4x
\]

### Final Answers:
1. \( (f \circ g)(x) = 4x - 12 \)
2. \( (g \circ f)(x) = 4x \)

Would you like a detailed explanation of the steps or further clarification?

### Related Questions:
1. What is the inverse of \( f(x) = 4x + 4 \) and \( g(x) = x - 4 \)?
2. How does the domain and range of \( (f \circ g)(x) \) differ from \( f(x) \) and \( g(x) \)?
3. Can the functions \( f(x) \) and \( g(x) \) be considered bijective? Why?
4. What happens when \( (f \circ g)(x) \) and \( (g \circ f)(x) \) are equal?
5. How does the order of composition affect the result of \( (f \circ g)(x) \) and \( (g \circ f)(x) \)?

**Tip:** Always substitute the inner function completely before simplifying in function composition to avoid errors!

If f(x) equals 4x plus 4 and g(x) equals x minus 4, find (f circle g)(x) and (g circle f)(x).

Learn how to solve the composition of functions (f ∘ g)(x) and (g ∘ f)(x) for f(x) = 4x + 4 and g(x) = x - 4. This detailed guide covers function composition concepts, linear function simplifications, and step-by-step calculations. Suitable for students in Grades 9-11.

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