Find (a) $f(g(x))$ , (b) $g(f(x))$ , and (c) $f(f(x))$ .

$f(x)=3x^{-1},\ g(x)=4x+8$

a. $f(g(x))=$ 

b.  $g(f(x))=$  

c.  $f(f(x))=$  

Question 2
State the domain of each composition.
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
(a) Domain: Response area except Response area

(b) Domain: Response area except Response area

(c) Domain: Response area

We are given two functions:
- $ f(x) = 3x^{-1} = \frac{3}{x} $
- $ g(x) = 4x + 8 $

Let's find each composition step by step.

### (a) $ f(g(x)) $
This means plugging $ g(x) $ into $ f(x) $. 
We have $ g(x) = 4x + 8 $, so:

\[
f(g(x)) = f(4x + 8) = \frac{3}{4x + 8}
\]

Thus, $ f(g(x)) = \frac{3}{4x + 8} $.

### (b) $ g(f(x)) $
This means plugging $ f(x) $ into $ g(x) $. 
We have $ f(x) = \frac{3}{x} $, so:

\[
g(f(x)) = g\left(\frac{3}{x}\right) = 4\left(\frac{3}{x}\right) + 8 = \frac{12}{x} + 8
\]

Thus, $ g(f(x)) = \frac{12}{x} + 8 $.

### (c) $ f(f(x)) $
This means plugging $ f(x) $ into itself. We know $ f(x) = \frac{3}{x} $, so:

\[
f(f(x)) = f\left(\frac{3}{x}\right) = \frac{3}{\frac{3}{x}} = \frac{3x}{3} = x
\]

Thus, $ f(f(x)) = x $.

---

### Domain Analysis

Now, let's determine the domain for each composition function:

- **(a) $ f(g(x)) = \frac{3}{4x + 8} $**: 
  The expression is undefined when the denominator equals zero, i.e., $ 4x + 8 = 0 $, which gives $ x = -2 $. Therefore, the domain of $ f(g(x)) $ is all real numbers **except $ x = -2 $**.

- **(b) $ g(f(x)) = \frac{12}{x} + 8 $**: 
  This expression is undefined when $ x = 0 $ (division by zero). Therefore, the domain of $ g(f(x)) $ is all real numbers **except $ x = 0 $**.

- **(c) $ f(f(x)) = x $**: 
  The domain of $ f(x) $ is $ x \neq 0 $ (since division by zero is undefined in $ f(x) $). Thus, the domain of $ f(f(x)) $ is all real numbers **except $ x = 0 $**.

---

### Final Answers

(a) $ f(g(x)) = \frac{3}{4x + 8} $  
(b) $ g(f(x)) = \frac{12}{x} + 8 $  
(c) $ f(f(x)) = x $

Domains:
- (a) Domain: All real numbers except $ x = -2 $
- (b) Domain: All real numbers except $ x = 0 $
- (c) Domain: All real numbers except $ x = 0 $

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Would you like further details or have any questions?

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### Relative Questions:
1. How do you determine the domain of a function?
2. What happens when you compose functions with restricted domains?
3. Can you explain how the composition of functions works step by step?
4. How would the domain of $ f(g(x)) $ change if $ g(x) $ was a different function?
5. What is the difference between $ f(g(x)) $ and $ g(f(x)) $ in terms of application?

### Tip:
When finding the domain of composite functions, always check where the individual functions are undefined.

Find (a) f(g(x)) , (b) g(f(x)) , and (c) f(f(x)). 
f(x)=3x^{-1}, g(x)=4x+8

(a) f(g(x))=
(b) g(f(x))=
(c) f(f(x))=

Question 2: State the domain of each composition.

Learn how to find the composition of two functions, f(g(x)), g(f(x)), and f(f(x)), using the given functions f(x) = 3/x and g(x) = 4x + 8. Explore step-by-step solutions and understand the domains of these composite functions. Suitable for students in Grades 10-12.

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