What should be the name of this compound containg both double and triple bond?
According to R-3.1.1 of IUPAC, Commission on Nomenclature of Organic Chemistry:

Numbers as low as possible are given to double and triple bonds as a set, even though this may at times give &quot;-yne&quot; a lower number than &quot;-ene&quot;. If a choice remains, preference for low locants is given to the double bonds. Only the lower locant for a multiple bond is cited except when the numerical difference between the two locants is greater than one, in which case the higher-numbered locant is cited in parentheses (see R-0.1.4)

Numbering from the right side of the molecule as seen in the question figure will assign the locants (2,4) to the double and triple bonds respectively.
On the other hand, following a similar numbering process from the left side of the molecule as seen in the diagram will yield (3,1) as the locants for the double and triple bonds respectively.
Now, as per IUPAC Recommendations 1:

P-14.3.5 Lowest set of locants
The lowest set of locants is defined as the set that, when compared term by term with other locant sets, each cited in order of increasing value, has the lowest term at the first point of difference; for example, the locant set ‘2,3,5,8’ is lower than ‘3,4,6,8’ and ‘2,4,5,7’.

Therefore, since (1,3) is a lower set of locants as compared to (2,4) as per the above definition, the IUPAC name of the compound should be pent-3-en-1-yne
Note that this molecule can also exhibit stereoisomerism by the virtue of it's double bond, so depending on whether the hydrogens are cis or trans to each other, the complete name of the molecule could be (3 Z)-pent-3-en-1-yne or (3​E)-pent-3-en-1-yne, respectively.