Find the Range of function
f(x) = `log_(e) (3x^(2) - 4 x + 5) ` is :

Options

A. `(log_e(11//3),oo)`
B. `[log_(e) (11//3 ) , oo)`
C. `(1 , oo)`
D. None of these

Answer

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

### Step 1: Identify the quadratic function
We need to analyze the expression inside the logarithm, which is \( g(x) = 3x^2 - 4x + 5 \).

### Step 2: Determine the nature of the quadratic function
The quadratic function \( g(x) = 3x^2 - 4x + 5 \) is a parabola that opens upwards because the coefficient of \( x^2 \) (which is 3) is positive.

### Step 3: Find the vertex of the quadratic function
The vertex of a parabola given by \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).
Here, \( a = 3 \) and \( b = -4 \).

\[ x = -\frac{-4}{2 \cdot 3} = \frac{4}{6} = \frac{2}{3} \]

### Step 4: Calculate the minimum value of \( g(x) \)
Now, substitute \( x = \frac{2}{3} \) back into \( g(x) \) to find the minimum value.

\[ g\left(\frac{2}{3}\right) = 3\left(\frac{2}{3}\right)^2 - 4\left(\frac{2}{3}\right) + 5 \]
\[ = 3 \cdot \frac{4}{9} - \frac{8}{3} + 5 \]
\[ = \frac{12}{9} - \frac{8}{3} + 5 \]
\[ = \frac{12}{9} - \frac{24}{9} + \frac{45}{9} \]
\[ = \frac{12 - 24 + 45}{9} = \frac{33}{9} = \frac{11}{3} \]

### Step 5: Determine the range of \( g(x) \)
Since the parabola opens upwards and the minimum value is \( \frac{11}{3} \), the range of \( g(x) \) is:

\[ \left[\frac{11}{3}, \infty\right) \]

### Step 6: Find the range of \( f(x) \)
Now, since \( f(x) = \log_e(g(x)) \), we need to find the range of \( f(x) \) based on the range of \( g(x) \).

The logarithmic function \( \log_e(x) \) is defined for \( x > 0 \) and is increasing. Therefore, we can find the range of \( f(x) \) by evaluating:

\[ \text{Range of } f(x) = \left[\log_e\left(\frac{11}{3}\right), \log_e(\infty)\right) \]

Since \( \log_e(\infty) = \infty \), the final range of \( f(x) \) is:

\[ \left[\log_e\left(\frac{11}{3}\right), \infty\right) \]

### Final Answer
The range of the function \( f(x) = \log_e(3x^2 - 4x + 5) \) is:

\[ \left[\log_e\left(\frac{11}{3}\right), \infty\right) \]