Question:
Let \(A=\{x\in\mathbf{R}: x\) is not a positive integer\(\}\). Define a function \(f:A\to\mathbf{R}\) as \(f(x)=\frac{2x}{x-1}\), then \(f\) is :
Let \(A=\{x\in\mathbf{R}: x\) is not a positive integer\(\}\). Define a function \(f:A\to\mathbf{R}\) as \(f(x)=\frac{2x}{x-1}\), then \(f\) is :
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Test \(f(a)=f(b)\) for injectivity, and solve \(y=f(x)\) for \(x\) to see which \(y\) are missed because \(x\) must not be a positive integer.
Updated On: Nov 5, 2026
- not injective
- neither injective nor surjective
- surjective but not injective
- injective but not surjective
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The Correct Option is D
Solution and Explanation
Injective: \(f(a)=f(b)\Rightarrow \frac{2a}{a-1}=\frac{2b}{b-1}\Rightarrow 2a(b-1)=2b(a-1)\Rightarrow -2a=-2b\Rightarrow a=b\). So \(f\) is one-one.
Onto: \(y=\frac{2x}{x-1}\Rightarrow x=\frac{y}{y-2}\) (so \(y\neq 2\)). For \(y=4\): \(x=\frac{4}{2}=2\), a positive integer, which is not in \(A\). So \(4\) has no pre-image in \(A\) (likewise \(y=2\) has none), hence \(f\) is not onto \(\mathbf{R}\).
Answer: (4) injective but not surjective
Onto: \(y=\frac{2x}{x-1}\Rightarrow x=\frac{y}{y-2}\) (so \(y\neq 2\)). For \(y=4\): \(x=\frac{4}{2}=2\), a positive integer, which is not in \(A\). So \(4\) has no pre-image in \(A\) (likewise \(y=2\) has none), hence \(f\) is not onto \(\mathbf{R}\).
Answer: (4) injective but not surjective
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