Match the following
List - I List - II
I) $f : R \to R$ defined by $f (x) = a x + b$ is           P) Injection but not surjection
II) $f : R \to R$ defined by $f (x) = [x]$ is           Q) Surjection but not Injection
III) $f : R \to [0, \infty)$ defined by $f (x) = |x|$ is           R) Bijection
IV) $f : N \to N$ defined by $f (x) = x^{3}$ is           S) Neither Injection nor Surjection

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$I \to R, II \to S, III \to Q, IV \to P$

$I \to S, II \to P, III \to Q, IV \to R$

$I \to Q, II \to S, III \to R, IV \to S$

$I \to P, II \to Q, III \to R, IV \to S$

answer is D.

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Detailed Solution

I) $f (x_{1}) = f (x_{2})$

$a x_{1} + b = a x_{2} + b$

$x_{1} = x_{2}$

$\Rightarrow f$ is one-one $y = a x + b$

$x = \frac{y - b}{a} \Rightarrow f$ is onto

$∴ f$ is a bijection

II) Different elements has same image.

Range is not equal to co-domain of f is neither injection nor surjection

III) Different elements have same image.

Surjection but not injection

IV) Different elements have same image

Range is not equal to co-domain.

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	List - I		List - II
I)	$f : R \to R$ defined by $f (x) = a x + b$ is	P)	Injection but not surjection
II)	$f : R \to R$ defined by $f (x) = [x]$ is	Q)	Surjection but not Injection
III)	$f : R \to [0, \infty)$ defined by $f (x) = \|x\|$ is	R)	Bijection
IV)	$f : N \to N$ defined by $f (x) = x^{3}$ is	S)	Neither Injection nor Surjection