In a bijective function is it necessary that all the elements of the domain correspond to a value in the range? Like for example can the following be a bijective function - 
If not then what type of function is it?
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Thanks for the clarification – Abhinandan Saha Mar 24 '21 at 03:53
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The domain of a function is the set of all possible inputs, which in the depicted case is ${1,2,3,4}. \qquad$ – Michael Hardy Mar 24 '21 at 03:55
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Bijection between which sets? As the comments and answers show, there's some ambiguity here. – saulspatz Mar 24 '21 at 03:57
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It's not a function at all – Righter Mar 24 '21 at 06:33
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As per the diagram, it's not a function. A function assigns to every value $x$ in domain (which from the diagram seems to be the set $\{1,2,3,4,5,6\}$) a unique value in Co-domain (which seems to be equal to range from the diagram). Therefore, in your case, $f:\{1,2,3,4,5,6\}\to \{a_1,a_2,a_3,a_4\}$ is not a function.
However, if you consider restriction of $\{1,2,3,4,5,6\}$ to $\{1,2,3,4\}$, then yes, $g:\{1,2,3,4\}\to \{a_1,a_2,a_3,a_4\}$ is a bijection in the way your diagram shows.
Koro
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This is a bijection from the set $\{1,2,3,4\}$ to the set $\{a_1,a_2,a_3,a_4\}.$
It is NOT a bijection from $\{1,2,3,4,5,6\}$ to $\{a_1,a_2,a_3,a_4\}.$
