`f(x)=log_(3)(5+4x-x^(2))`. find the range of f(x).

Answer

To find the range of the function \( f(x) = \log_{3}(5 + 4x - x^2) \), we need to analyze the expression inside the logarithm, which is \( 5 + 4x - x^2 \).

### Step 1: Rewrite the quadratic expression
We start by rewriting the quadratic expression \( 5 + 4x - x^2 \) in a standard form. We can rearrange it as:
\[ -x^2 + 4x + 5 \]
To make it easier to analyze, we can factor out a negative sign:
\[ -(x^2 - 4x - 5) \]

### Step 2: Complete the square
Next, we complete the square for the quadratic \( x^2 - 4x - 5 \):
\[ x^2 - 4x = (x - 2)^2 - 4 \]
Thus,
\[ -(x^2 - 4x - 5) = -((x - 2)^2 - 4 - 5) = -((x - 2)^2 - 9) = 9 - (x - 2)^2 \]

### Step 3: Determine the maximum value
The expression \( 9 - (x - 2)^2 \) reaches its maximum value when \( (x - 2)^2 = 0 \), which occurs at \( x = 2 \):
\[ 5 + 4(2) - (2)^2 = 9 \]
Thus, the maximum value of \( 5 + 4x - x^2 \) is \( 9 \).

### Step 4: Find the range of the logarithmic function
Since \( f(x) = \log_{3}(5 + 4x - x^2) \), we need to find the values of \( f(x) \) when \( 5 + 4x - x^2 \) is in the valid range for the logarithm, which is \( (0, 9] \).

### Step 5: Determine the limits for the logarithm
- As \( 5 + 4x - x^2 \) approaches \( 0 \), \( f(x) \) approaches \( -\infty \).
- When \( 5 + 4x - x^2 = 9 \), we calculate:
\[ f(x) = \log_{3}(9) = 2 \]

### Conclusion
Thus, the range of \( f(x) \) is:
\[ (-\infty, 2] \]