Let f, g: N - {1} rightarrow N be functions d | vedclass

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(D) Given $f(a) = \alpha$,where $\alpha$ is the maximum power of a prime $p$ such that $p^{\alpha}$ divides $a$. This is the exponent of the highest p

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neither one-one nor onto

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(D) Given $f(a) = \alpha$,where $\alpha$ is the maximum power of a prime $p$ such that $p^{\alpha}$ divides $a$. This is the exponent of the highest prime power dividing $a$.Let $h(a) = (f + g)(a) = f(a) + a + 1$.Calculate values for some $a \in N - \{1\}$:$h(2) = f(2) + 2 + 1 = 1 + 2 + 1 = 4$$h(3) = f(3) + 3 + 1 = 1 + 3 + 1 = 5$$h(4) = f(4) + 4 + 1 = 2 + 4 + 1 = 7$$h(5) = f(5) + 5 + 1 = 1 + 5 + 1 = 7$Since $h(4) = h(5) = 7$ for $4 
eq 5$,the function is not one-one.Also,the range of $h(a)$ does not include $1, 2, 3, 6, \dots$ (e.g.,$h(a) \ge 4$ for all $a \ge 2$),so it is not onto.Therefore,the function is neither one-one nor onto.

$(A)$ $f: R \rightarrow R$ is such that $f(x)=px+q$ $(p \neq 0)$,$\forall x \in R$	$I.$ $f$ is neither one-one nor onto
$(B)$ $f: R \rightarrow R^{+} \cup\{0\}$ is such that $f(x)=x^2$,$\forall x \in R$	$II.$ $f$ is both one-one and onto
$(C)$ $f: N \rightarrow N$ is such that $f(n)=n^2+2n+3$,$\forall n \in N$	$III.$ $f$ is one-one but not onto
$(D)$ $f: R \rightarrow R$ is such that $f(x)=2(\cos ^2 5x+\sin ^2 5x)$ $\forall x \in R$	$IV.$ $f$ is onto but not one-one
	$V.$ $f$ is a constant function and also a bijection

Let $f, g: N - \{1\} \rightarrow N$ be functions defined by $f(a) = \alpha$,where $\alpha$ is the maximum of the powers of those primes $p$ such that $p^{\alpha}$ divides $a$,and $g(a) = a + 1$,for all $a \in N - \{1\}$. Then,the function $f + g$ is.

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