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: Set the function equal to \( y \)
Let \( f(x) = y \). Therefore, we have:
\[ y = \log_3(5 + 4x - x^2) \]

### Step 2: Convert the logarithmic equation to an exponential form
Using the property of logarithms, we can rewrite the equation as:
\[ 5 + 4x - x^2 = 3^y \]

### Step 3: Rearrange the equation
Rearranging gives us a quadratic equation in terms of \( x \):
\[ -x^2 + 4x + (5 - 3^y) = 0 \]
or
\[ x^2 - 4x + (3^y - 5) = 0 \]

### Step 4: Apply the discriminant condition
For \( x \) to have real solutions, the discriminant of this quadratic equation must be non-negative. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot (3^y - 5) \]
Calculating this gives:
\[ D = 16 - 4(3^y - 5) = 16 - 4 \cdot 3^y + 20 = 36 - 4 \cdot 3^y \]

### Step 5: Set the discriminant greater than or equal to zero
To ensure real solutions for \( x \), we need:
\[ 36 - 4 \cdot 3^y \geq 0 \]
This simplifies to:
\[ 36 \geq 4 \cdot 3^y \]
Dividing both sides by 4 gives:
\[ 9 \geq 3^y \]

### Step 6: Rewrite the inequality
We can express 9 as \( 3^2 \):
\[ 3^y \leq 3^2 \]
Since the bases are the same, we can compare the exponents:
\[ y \leq 2 \]

### Step 7: Determine the range of \( f(x) \)
Since \( y \) can take any value less than or equal to 2, we conclude that the range of the function \( f(x) \) is:
\[ (-\infty, 2] \]

### Final Answer
The range of the function \( f(x) = \log_3(5 + 4x - x^2) \) is:
\[ (-\infty, 2] \]

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