https://file.aiquickdraw.com/mathbot/file/17272739338123m2f1wax.pdf

Here are the solutions to all the questions from the document on **functions, compositions, and inverse functions**:

### Soal No. 1:
Given:  
\( f(x) = 2x^4 - 3x^3 + 12x - 5 \)  
\( g(x) = 10x^4 + 7x^3 - 11x^2 + 17x + 1 \)  

To find \( (f + g)(x) \):  
\[
(f + g)(x) = f(x) + g(x)
\]
\[
(f + g)(x) = (2x^4 - 3x^3 + 12x - 5) + (10x^4 + 7x^3 - 11x^2 + 17x + 1)
\]
\[
= 12x^4 + 4x^3 - 11x^2 + 29x - 4
\]

### Soal No. 2:
Given:  
\( f(x) = 6x^3 - 8x^2 + 9x - 15 \)  
\( g(x) = 12x^3 + 2x^2 + 3x \)  

To find \( (f - g)(x) \):  
\[
(f - g)(x) = f(x) - g(x)
\]
\[
(f - g)(x) = (6x^3 - 8x^2 + 9x - 15) - (12x^3 + 2x^2 + 3x)
\]
\[
= -6x^3 - 10x^2 + 6x - 15
\]

### Soal No. 3:
Given:  
\( f(x) = 3x^2 + 7x - 5 \)  
\( g(x) = 4x^2 - 8x \)  

To find \( (f \cdot g)(x) \):  
\[
(f \cdot g)(x) = f(x) \cdot g(x)
\]
Multiply the two functions:
\[
(f \cdot g)(x) = (3x^2 + 7x - 5)(4x^2 - 8x)
\]
Expanding:
\[
= 12x^4 - 24x^3 + 28x^3 - 56x^2 - 20x^2 + 40x
\]
\[
= 12x^4 + 4x^3 - 76x^2 + 40x
\]

### Soal No. 4:
Given:  
\( f(x) = 9x^2 + 18x \)  
\( g(x) = 5x^2 - 24x \)  

To find \( \frac{f(x)}{g(x)} \):  
\[
\frac{f(x)}{g(x)} = \frac{9x^2 + 18x}{5x^2 - 24x}
\]
Factor both the numerator and denominator:
\[
\frac{f(x)}{g(x)} = \frac{9x(x + 2)}{x(5x - 24)}
\]
Simplifying:
\[
\frac{f(x)}{g(x)} = \frac{9(x + 2)}{5x - 24}
\]

### Soal No. 5:
Given:  
\( f(x) = 2x^2 + 13x + 21 \)  

To find \( f(-8) \):  
\[
f(-8) = 2(-8)^2 + 13(-8) + 21
\]
\[
= 2(64) - 104 + 21
\]
\[
= 128 - 104 + 21 = 45
\]

Would you like to explore any particular solution in detail or have additional questions? 

Here are 5 related questions to practice:

1. How do you derive the inverse of a function?
2. What is the difference between function addition and function multiplication?
3. How is the domain affected in a function composition?
4. How do you compute the value of a composite function, such as \( f(g(x)) \)?
5. What are the key properties of polynomial functions in algebra?

**Tip**: Always double-check the domain and range when working with composite or inverse functions to ensure the operations are valid!

Find detailed solutions to math problems involving functions, compositions, and inverse functions, covering topics such as polynomial operations, function addition, and composition rules. Perfect for high school students (Grades 10-12).

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