The range of the function `f(x) = log_(3) (5+4x - x^(2))`, is

Options

A. `(0, 2 ]`
B. `(-oo, 2]`
C. `(0, 9]`
D. none of these

Explanation

To find the range of the function \( f(x) = \log_{3}(5 + 4x - x^{2}) \), we will follow these steps:

### Step 1: Identify the expression inside the logarithm
We need to analyze the expression \( 5 + 4x - x^{2} \). This is a quadratic function, and we can rewrite it in standard form:
\[ f(x) = -x^{2} + 4x + 5 \]

### Step 2: Determine the nature of the quadratic
The coefficient of \( x^{2} \) is negative, which indicates that the parabola opens downwards. This means it has a maximum point.

### Step 3: Find the vertex of the quadratic
To find the maximum value, we can use the vertex formula for a quadratic \( ax^{2} + bx + c \), where the x-coordinate of the vertex is given by:
\[ x = -\frac{b}{2a} \]
Here, \( a = -1 \) and \( b = 4 \):
\[ x = -\frac{4}{2 \times -1} = 2 \]

### Step 4: Calculate the maximum value of the quadratic
Now, we substitute \( x = 2 \) back into the quadratic to find the maximum value:
\[ f(2) = 5 + 4(2) - (2)^{2} = 5 + 8 - 4 = 9 \]

### Step 5: Determine the minimum value of the quadratic
Next, we need to find the values of \( x \) for which \( 5 + 4x - x^{2} > 0 \):
\[ 5 + 4x - x^{2} = 0 \]
Rearranging gives:
\[ -x^{2} + 4x + 5 = 0 \]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \):
Here, \( a = -1, b = 4, c = 5 \):
\[ x = \frac{-4 \pm \sqrt{4^{2} - 4(-1)(5)}}{2(-1)} = \frac{-4 \pm \sqrt{16 + 20}}{-2} = \frac{-4 \pm \sqrt{36}}{-2} \]
\[ = \frac{-4 \pm 6}{-2} \]
Calculating the two roots:
1. \( x = \frac{2}{-2} = -1 \)
2. \( x = \frac{-10}{-2} = 5 \)

Thus, the quadratic \( 5 + 4x - x^{2} \) is positive between the roots \( x = -1 \) and \( x = 5 \).

### Step 6: Determine the range of \( f(x) \)
Since \( 5 + 4x - x^{2} \) is positive in the interval \( (-1, 5) \), the logarithmic function \( f(x) \) will take values from \( \log_{3}(0) \) (which approaches \(-\infty\)) to \( \log_{3}(9) \) (which is \( 2 \)):
\[ \text{Range of } f(x) = (-\infty, 2) \]

### Final Answer
The range of the function \( f(x) = \log_{3}(5 + 4x - x^{2}) \) is \( (-\infty, 2) \).

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