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The image you uploaded presents a table with functions, and the task is to identify the type of function and the type of relation for each function listed.

Here’s a breakdown of each function:

1. **Function:** \( f(x) = 4x^3 - qx^2 \)
   - **Type of Function:** This is a **polynomial function** since it is a sum of terms where the variables have whole number exponents (3 and 2).
   - **Type of Relation:** The relation is **many-to-one**, as a polynomial function maps multiple values of \(x\) to a single value of \(f(x)\) in general.

2. **Function:** \( p(q) = 5 \log_3 9 \)
   - **Type of Function:** This is a **constant function** because \( \log_3 9 \) simplifies to 2, making the entire function equal to a constant value \(5 \times 2 = 10\).
   - **Type of Relation:** The relation is **one-to-one** as each input corresponds to a single output (in this case, the constant output 10).

3. **Function:** \( k(t) = \frac{4r - mt}{g} \)
   - **Type of Function:** This is a **rational function** because it is the ratio of two expressions involving variables.
   - **Type of Relation:** The relation is **many-to-one**, as each value of \( t \) typically corresponds to one value of \( k(t) \).

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Would you like more detailed explanations for any of these, or any further assistance with related concepts?

Here are 5 questions to expand your understanding:
1. What is the general form of a polynomial function, and how can you identify the degree?
2. How do you determine if a function is one-to-one or many-to-one?
3. Can you explain what makes a function rational, and what the denominator signifies in rational functions?
4. How can logarithmic functions be identified, and what types of functions do they belong to?
5. How does the behavior of polynomial functions differ from rational functions in terms of graph shape and behavior at infinity?

**Tip:** Always check the degree of the polynomial and the form of the terms to identify the type of polynomial function.

Copy and complete the table below by identifying the type of function and the type of relation for each function.

Learn how to identify different types of functions including polynomial, constant, rational, and logarithmic functions. This problem explains the types of relations such as many-to-one and one-to-one relations for each function. Suitable for students in Grades 10-12.

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