The domain and range of the function `f (x) = ""^(7-x)P_(x-3)`are ....... and ......
domain

Options

A. `[3.4]`
B. `[4,5]`
C. `[3,4,5]`
D. None

Answer

To find the domain and range of the function \( f(x) = (7-x)(x-3) \), we will follow these steps:

### Step 1: Identify the conditions for the function to be defined
The function \( f(x) = (7-x)(x-3) \) is a polynomial function, which is defined for all real numbers. However, we need to consider any restrictions that may arise from the components of the function.

### Step 2: Set up inequalities based on the components
1. The term \( 7 - x \) must be greater than or equal to 0:
\[ 7 - x \geq 0 \implies x \leq 7 \]

2. The term \( x - 3 \) must be greater than or equal to 0:
\[ x - 3 \geq 0 \implies x \geq 3 \]

### Step 3: Combine the inequalities
From the inequalities derived:
- \( x \leq 7 \)
- \( x \geq 3 \)

We can combine these inequalities to find the domain:
\[ 3 \leq x \leq 7 \]

### Step 4: Determine the range of the function
To find the range, we need to analyze the function within the domain \( [3, 7] \).

1. Evaluate the function at the endpoints of the interval:
- At \( x = 3 \):
\[ f(3) = (7 - 3)(3 - 3) = 4 \cdot 0 = 0 \]
- At \( x = 7 \):
\[ f(7) = (7 - 7)(7 - 3) = 0 \cdot 4 = 0 \]

2. Check the value of the function at the midpoint \( x = 5 \):
\[ f(5) = (7 - 5)(5 - 3) = 2 \cdot 2 = 4 \]

### Step 5: Analyze the function's behavior
Since \( f(x) \) is a quadratic function that opens downwards (as the leading coefficient is negative), the maximum value occurs at \( x = 5 \) with \( f(5) = 4 \). The function reaches a minimum value of 0 at both endpoints \( x = 3 \) and \( x = 7 \).

### Conclusion
Thus, the domain and range of the function \( f(x) \) are:
- **Domain**: \( [3, 7] \)
- **Range**: \( [0, 4] \)

### Final Answer
The domain and range of the function \( f(x) = (7-x)(x-3) \) are \( [3, 7] \) and \( [0, 4] \) respectively.

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