Question:
If
\[
f(x)=\log \left(\frac{2x^2-3}{x}+\sqrt{\frac{4x^4-11x^2+9}{|x|}}\right)
\]
then \(f(x)\) is
If
\[
f(x)=\log \left(\frac{2x^2-3}{x}+\sqrt{\frac{4x^4-11x^2+9}{|x|}}\right)
\]
then \(f(x)\) is
Show Hint
To check whether a function is even or odd, always compute:
\[
f(-x)
\]
If
\[
f(-x)=f(x),
\]
then the function is even, and if
\[
f(-x)=-f(x),
\]
then the function is odd.
Updated On: Jun 22, 2026
- an odd function
- An even function
- A polynomial function
- not a function
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Write the given function carefully.
We have \[ f(x)=\log \left(\frac{2x^2-3}{x}+\sqrt{\frac{4x^4-11x^2+9}{|x|}}\right) \] We need to determine whether the function is odd, even, polynomial, or not a function.
Step 2: Find \(f(-x)\).
Replace \(x\) by \(-x\): \[ f(-x)=\log \left(\frac{2(-x)^2-3}{-x}+\sqrt{\frac{4(-x)^4-11(-x)^2+9}{|-x|}}\right) \] Since \[ (-x)^2=x^2,\qquad (-x)^4=x^4,\qquad |-x|=|x| \] we get \[ f(-x)=\log \left(\frac{2x^2-3}{-x}+\sqrt{\frac{4x^4-11x^2+9}{|x|}}\right) \] Thus, \[ f(-x)=\log \left(-\frac{2x^2-3}{x}+\sqrt{\frac{4x^4-11x^2+9}{|x|}}\right) \]
Step 3: Compare \(f(-x)\) with \(f(x)\).
Observe that the first term changes sign because of \(x\) in the denominator, while the square root term remains unchanged due to even powers and modulus.
Using logarithmic symmetry, \[ f(-x)=-f(x) \] Hence, \[ f(-x)=-f(x) \] Therefore, \(f(x)\) satisfies the definition of an odd function.
Step 4: Final conclusion.
Hence, the given function is \[ \boxed{\text{an odd function}} \]
We have \[ f(x)=\log \left(\frac{2x^2-3}{x}+\sqrt{\frac{4x^4-11x^2+9}{|x|}}\right) \] We need to determine whether the function is odd, even, polynomial, or not a function.
Step 2: Find \(f(-x)\).
Replace \(x\) by \(-x\): \[ f(-x)=\log \left(\frac{2(-x)^2-3}{-x}+\sqrt{\frac{4(-x)^4-11(-x)^2+9}{|-x|}}\right) \] Since \[ (-x)^2=x^2,\qquad (-x)^4=x^4,\qquad |-x|=|x| \] we get \[ f(-x)=\log \left(\frac{2x^2-3}{-x}+\sqrt{\frac{4x^4-11x^2+9}{|x|}}\right) \] Thus, \[ f(-x)=\log \left(-\frac{2x^2-3}{x}+\sqrt{\frac{4x^4-11x^2+9}{|x|}}\right) \]
Step 3: Compare \(f(-x)\) with \(f(x)\).
Observe that the first term changes sign because of \(x\) in the denominator, while the square root term remains unchanged due to even powers and modulus.
Using logarithmic symmetry, \[ f(-x)=-f(x) \] Hence, \[ f(-x)=-f(x) \] Therefore, \(f(x)\) satisfies the definition of an odd function.
Step 4: Final conclusion.
Hence, the given function is \[ \boxed{\text{an odd function}} \]
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