If the Marginal revenue function for output x is given by 𝑀𝑅 = 
[4/(2𝑥+3)2
]−1, find the total revenue function and the demand 
function.

To find the total revenue function \( R(x) \) and the demand function \( p(x) \) from the marginal revenue function \( MR(x) \), we'll follow these steps:

### 1. **Finding the Total Revenue Function \( R(x) \):**

The marginal revenue \( MR(x) \) is the derivative of the total revenue \( R(x) \) with respect to the output \( x \):

\[
MR(x) = \frac{dR(x)}{dx}
\]

Given \( MR(x) = \left[\frac{4}{(2x + 3)^2}\right] - 1 \), we integrate \( MR(x) \) with respect to \( x \) to find \( R(x) \):

\[
R(x) = \int \left[\frac{4}{(2x + 3)^2} - 1\right] dx
\]

We can split the integral:

\[
R(x) = \int \frac{4}{(2x + 3)^2} dx - \int 1 \, dx
\]

#### **First Integral:**

Let \( u = 2x + 3 \), then \( du = 2 \, dx \) or \( dx = \frac{du}{2} \).

\[
\int \frac{4}{(2x + 3)^2} dx = \int \frac{4}{u^2} \cdot \frac{du}{2} = 2 \int u^{-2} du
\]

The integral of \( u^{-2} \) is \( -u^{-1} \), so:

\[
2 \int u^{-2} du = -\frac{2}{u} = -\frac{2}{2x + 3}
\]

#### **Second Integral:**

\[
\int 1 \, dx = x
\]

Putting it all together:

\[
R(x) = -\frac{2}{2x + 3} - x + C
\]

Where \( C \) is the constant of integration.

### 2. **Finding the Demand Function \( p(x) \):**

The total revenue function is also related to the price (demand) function \( p(x) \) by:

\[
R(x) = p(x) \cdot x
\]

Given \( R(x) = -\frac{2}{2x + 3} - x + C \), and using the relationship \( R(x) = p(x) \cdot x \), we solve for \( p(x) \):

\[
p(x) = \frac{R(x)}{x} = \frac{-\frac{2}{2x + 3} - x + C}{x}
\]

This expression represents the demand function \( p(x) \).

### Final Answer:

- **Total Revenue Function**: \( R(x) = -\frac{2}{2x + 3} - x + C \)
- **Demand Function**: \( p(x) = \frac{-\frac{2}{2x + 3} - x + C}{x} \)

Would you like more details on any of these steps? 

### Related Questions:
1. How do you determine the constant \( C \) in the total revenue function?
2. What is the economic interpretation of the demand function?
3. How does the marginal revenue function affect the shape of the total revenue curve?
4. Can the demand function \( p(x) \) be simplified further?
5. What happens to the total revenue if output increases significantly?

### Tip:
Always check if the function you've integrated correctly represents the original marginal function by differentiating the total revenue function again.

If the Marginal revenue function for output x is given by MR = [4/(2x+3)^2]−1, find the total revenue function and the demand function.

This problem involves finding the total revenue and demand functions given a marginal revenue function. Explore the detailed solution including integration techniques and the application of the Fundamental Theorem of Calculus. Suitable for students in Grades 11-12.

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