https://file.aiquickdraw.com/mathbot/file/1728360894762w6iu80w0.jpg

do questions 3,6,9,12,15,18 and 21 from the image

Here are the solutions for the specified questions from the image:

### Question 3
**\( f(x) = 3 \cdot 2^x \)**

- **Power function check**: A power function has the form \( f(x) = kx^a \). This equation does not follow that form because the exponent \( x \) is applied to a constant (2) rather than the variable \( x \). Thus, **this is not a power function**.

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### Question 6
**\( KE(v) = \frac{1}{2} k v^5 \)**

- **Power function check**: This is a power function because it has the form \( KE(v) = k v^a \), where \( k = \frac{1}{2} \) and the exponent \( a = 5 \).
- **Constant of variation**: The constant of variation is \( \frac{1}{2} k \).

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### Question 9
**\( l = d^2 \)**

- **Power function check**: This is a power function because it has the form \( l = kd^a \), where \( k = 1 \) and the exponent \( a = 2 \).
- **Constant of variation**: The constant of variation is \( 1 \).

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### Question 12
**\( f(x) = 3x^5 \)**

- **Monomial function check**: A monomial function has the form \( f(x) = ax^n \). This is a monomial function because it follows that form, where \( a = 3 \) and \( n = 5 \).
- **Degree**: The degree is \( 5 \).
- **Leading coefficient**: The leading coefficient is \( 3 \).

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### Question 15
**\( S = 9 \pi r^3 \)**

- **Monomial function check**: This is a monomial function since it has the form \( S = a r^n \), where \( a = 9 \pi \) and \( n = 3 \).
- **Degree**: The degree is \( 3 \).
- **Leading coefficient**: The leading coefficient is \( 9 \pi \).

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### Question 18
**The volume \( V \) of a circular cylinder with fixed height is proportional to the square of its radius \( r \).**

- **Power function equation**: The relationship described is \( V = kr^2 \), where \( k \) is a constant of variation based on the height of the cylinder.

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### Question 21
**The energy \( E \) produced in a nuclear reaction is proportional to the mass \( m \), with the constant of variation being \( c^2 \), the square of the speed of light.**

- **Power function equation**: The equation is \( E = c^2 m \).

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Would you like more details or have any questions on these solutions?

Here are 5 related questions for further practice:
1. What is the general form of a power function?
2. How can you determine whether an equation is a power function?
3. What is the degree and leading coefficient of \( f(x) = 7x^8 \)?
4. How does the constant of variation affect the behavior of a power function?
5. How would you rewrite \( f(x) = x^{\frac{1}{3}} \) as a power function?

**Tip**: Always check if the exponent is applied to the variable to determine if it's a power function!

Detailed solutions for selected problems involving power and monomial functions, constant of variation, and equations for energy and volume of a cylinder. This content is suitable for high school students in Grades 9-12.

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