Question:
The range of the function $f(x)=\log_{e}(4x^{2}-4x+1)$ where $x \ne \frac{1}{2}$ is ________.
The range of the function $f(x)=\log_{e}(4x^{2}-4x+1)$ where $x \ne \frac{1}{2}$ is ________.
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Logarithmic functions typically have a range of all real numbers.
Updated On: Jun 26, 2026
- $(-\infty,0)$
- $[0,\infty)$
- $(0,\infty)$
- $(-\infty,0]$
- $(-\infty,\infty)$
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The Correct Option is
Solution and Explanation
Step 1: Concept
Range refers to the set of all possible output values of the function.
Step 2: Meaning
$f(x) = \log_{e}((2x-1)^2)$. Since $x \ne \frac{1}{2}$, $(2x-1)^2 > 0$.
Step 3: Analysis
The term $(2x-1)^2$ can take any value in $(0, \infty)$.
Step 4: Conclusion
The logarithm of values from $(0, \infty)$ spans the entire real line $(-\infty, \infty)$. Final Answer: (E)
Range refers to the set of all possible output values of the function.
Step 2: Meaning
$f(x) = \log_{e}((2x-1)^2)$. Since $x \ne \frac{1}{2}$, $(2x-1)^2 > 0$.
Step 3: Analysis
The term $(2x-1)^2$ can take any value in $(0, \infty)$.
Step 4: Conclusion
The logarithm of values from $(0, \infty)$ spans the entire real line $(-\infty, \infty)$. Final Answer: (E)
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