Explain, using these theorems, why the function is continuous at every number in its domain.
M(x) = 

1 + 
3
x
M(x) is a polynomial, so it is continuous at every number in its domain.
M(x) is a rational function, so it is continuous at every number in its domain.    
M(x) is an exponential function, so it is continuous at every number in its domain.
M(x) is a composition of functions that are continuous on the domain of M(x), so it is continuous at every number in its domain.
M(x) is not continuous at every number in its domain.
Correct: Your answer is correct.
State the domain. (Enter your answer using interval notation.)
  
Incorrect: Your answer is incorrect.

To explain why the function \( M(x) = 1 + \frac{3}{x} \) is continuous at every number in its domain, we can use several mathematical theorems and concepts about different types of functions. Let’s break it down step by step using the provided theorems.

### 1. **M(x) is a rational function, so it is continuous at every number in its domain.**
   - The function \( M(x) = 1 + \frac{3}{x} \) is a **rational function** because it involves a ratio of two polynomials. Specifically, the term \( \frac{3}{x} \) is a rational expression. 
   - A key property of rational functions is that they are continuous **everywhere in their domain**. The only points where a rational function might not be continuous are where the denominator equals zero.
   - In this case, the denominator is \( x \), so the function is not defined at \( x = 0 \). Therefore, \( M(x) \) is continuous for all \( x \neq 0 \).

### 2. **M(x) is a polynomial, so it is continuous at every number in its domain.**
   - This statement doesn't fully apply because \( M(x) \) itself is not a polynomial (it has a rational term \( \frac{3}{x} \), not just powers of \( x \)). However, polynomials themselves are continuous at every number in their domain, and the constant \( 1 \) (which is a polynomial) is continuous everywhere.

### 3. **M(x) is an exponential function, so it is continuous at every number in its domain.**
   - This statement is incorrect in the case of \( M(x) \), because \( M(x) \) is not an exponential function (it doesn't have the form \( a^x \) where \( a \) is a constant base). So this theorem is not applicable here.

### 4. **M(x) is a composition of functions that are continuous on the domain of M(x), so it is continuous at every number in its domain.**
   - This statement **does** apply to \( M(x) \). The function \( M(x) \) can be seen as a composition of simpler functions: the constant function \( 1 \), the rational function \( \frac{3}{x} \), and the addition operation. Since each of these individual functions is continuous on their respective domains, and compositions of continuous functions are continuous, \( M(x) \) is continuous everywhere in its domain.

### 5. **M(x) is not continuous at every number in its domain.**
   - This statement is incorrect. As established, \( M(x) \) is continuous at all points in its domain. The only point of concern is \( x = 0 \), but this point is not part of the domain, so it doesn't affect continuity.

### **State the domain**

The function \( M(x) = 1 + \frac{3}{x} \) is undefined when \( x = 0 \) because division by zero is not allowed. Therefore, the domain of \( M(x) \) is all real numbers except 0. In interval notation, this is written as:

\[
\boxed{(-\infty, 0) \cup (0, \infty)}
\]

This means that \( M(x) \) is defined for all real numbers except zero, and it is continuous everywhere in its domain.

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Do you need further details or have any questions?

Here are 5 questions to expand your understanding:
1. What are the general conditions for a rational function to be discontinuous?
2. Why are polynomials continuous everywhere by definition?
3. How does the composition of continuous functions remain continuous?
4. Can a function have discontinuities even if its components are continuous? If yes, under what circumstances?
5. How would you describe the domain of a more complex rational function, like \( \frac{2x + 3}{x^2 - 4} \)?

**Tip:** When determining the domain of a function involving fractions, always check when the denominator becomes zero, as this indicates points of discontinuity.

Explain, using these theorems, why the function is continuous at every number in its domain. M(x) = 1 + 3/x. M(x) is a polynomial, so it is continuous at every number in its domain. M(x) is a rational function, so it is continuous at every number in its domain. M(x) is an exponential function, so it is continuous at every number in its domain. M(x) is a composition of functions that are continuous on the domain of M(x), so it is continuous at every number in its domain. M(x) is not continuous at every number in its domain. State the domain. (Enter your answer using interval notation.)

Understand why the function M(x) = 1 + 3/x is continuous at every number in its domain using key mathematical theorems. Explore the concepts of rational functions, continuity, and domain determination, with detailed explanations and examples. Ideal for students in Grades 10-12.

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