Find 
(g ∘ f)(x)
 and 
(f ∘ g)(x)
 for the given functions f and g.
f(x) = 

x + 5
,    g(x) = 
1
x
(g ∘ f)(x)	 = 	
  

(f ∘ g)(x)	 = 	
  



To find \((g \circ f)(x)\) and \((f \circ g)(x)\), we need to compute the compositions of the functions \(f(x) = x + 5\) and \(g(x) = \frac{1}{x}\).

### 1. Finding \((g \circ f)(x)\):
\[
(g \circ f)(x) = g(f(x))
\]
This means we substitute \(f(x)\) into \(g(x)\).

1. Start with \(f(x) = x + 5\).
2. Substitute \(x + 5\) into \(g(x) = \frac{1}{x}\):
   \[
   g(f(x)) = g(x + 5) = \frac{1}{x + 5}
   \]

Therefore, 
\[
(g \circ f)(x) = \frac{1}{x + 5}
\]

### 2. Finding \((f \circ g)(x)\):
\[
(f \circ g)(x) = f(g(x))
\]
This means we substitute \(g(x)\) into \(f(x)\).

1. Start with \(g(x) = \frac{1}{x}\).
2. Substitute \(\frac{1}{x}\) into \(f(x) = x + 5\):
   \[
   f(g(x)) = f\left(\frac{1}{x}\right) = \frac{1}{x} + 5
   \]

Therefore, 
\[
(f \circ g)(x) = \frac{1}{x} + 5
\]

### Summary:
\[
(g \circ f)(x) = \frac{1}{x + 5}
\]

\[
(f \circ g)(x) = \frac{1}{x} + 5
\]

Would you like more details or have any questions?

Here are 5 related questions:

1. What are the domains of \( (g \circ f)(x) \) and \( (f \circ g)(x) \)?
2. How do you find the inverse of a composition of two functions?
3. What is the range of \( (g \circ f)(x) \)?
4. Can you find \( (f \circ f)(x) \) and \( (g \circ g)(x) \)?
5. How do you verify if two functions are inverses of each other?

**Tip:** When working with function compositions, always check the domains of the individual functions and the composed function to ensure that all values are valid.

Find (g ∘ f)(x) and (f ∘ g)(x) for the given functions f and g. f(x) = x + 5, g(x) = 1/x

Discover how to find the function compositions (g ∘ f)(x) = 1/(x + 5) and (f ∘ g)(x) = 1/x + 5. This algebra problem explores function composition and domain restrictions, ideal for students in Grades 9-12.

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