Which of the following statements is true?

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(A) Let us analyze each option:$A$. If $A$ is a finite set with $n$ elements,a function $f: A 	o A$ is one-one if and only if it is onto (by the Pige

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If the number of elements in set $A$ is finite,such that $f : A 	o A$ is a one-one function,then $f$ is necessarily onto.

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(A) Let us analyze each option:$A$. If $A$ is a finite set with $n$ elements,a function $f: A 	o A$ is one-one if and only if it is onto (by the Pigeonhole Principle). Thus,this statement is true.$B$. By the Intermediate Value Theorem,if a continuous function changes sign between $x_1$ and $x_2$,there is at least one root. However,it does not guarantee an odd number of roots; there could be an even number of roots if the function crosses the axis multiple times.$C$. If $A$ is an infinite set (e.g.,$f: \mathbb{N} 	o \mathbb{N}$ defined by $f(x) = x + 1$),the function is one-one but not onto. Thus,this is false.$D$. $A$ local maximum and a global minimum cannot occur at the same point unless the function is constant,which contradicts the definition of local maxima/minima in standard contexts. Thus,this is false.Therefore,the correct statement is $A$.

$A$. $f: R \rightarrow R$ defined by $f(x) = \cos(112x - 37)$	$I$. Injection but not surjection
$B$. $f: A \rightarrow B$ defined by $f(x) = x\|x\|$ when $A = [-2, 2]$ and $B = [-4, 4]$	$II$. Surjection but not injection
$C$. $f: R \rightarrow R$ defined by $f(x) = (x-2)(x-3)(x-5)$	$III$. Bijection
$D$. $f: N \rightarrow N$ defined by $f(n) = n+1$	$IV$. Neither injection nor surjection
	$V$. Composite function

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