Question

The range of the function $f(x)=\logₑ(3x²-4x+5)$ is ________.

• A. $(-\infty, \log_{e}\frac{11}{3}]$
• B. $[\log_{e}\frac{11}{3}, \infty)$
• C. $[-\log_{e}\frac{11}{3}, \log_{e}\frac{11}{3}]$
• D. None of the above

Hint:

The range of the function $f(x)=\loge(3x-4x+5)$ is ____.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Find the range of the quadratic expression $3x^2 - 4x + 5$ first.
Step 2: Quadratic Analysis
The minimum value of a quadratic $ax^2 + bx + c$ with $a>0$ is $\frac{4ac - b^2}{4a}$. Minimum value $= \frac{4(3)(5) - (-4)^2}{4(3)} = \frac{60 - 16}{12} = \frac{44}{12} = \frac{11}{3}$.
Step 3: Logarithmic Range
Since the minimum value of the inner expression is $11/3$ and it increases to infinity, the range of the log function is $[\log_{e}\frac{11}{3}, \infty)$.
Hence, the Answer is: (B)