Three consonants carry one idea: placing something where it can be seen. From there the root fans out two ways at once: to show a thing (put it on display) and to represent someone (stand in their place). The same letters give you a theatre show, a computer screen, an art installation, and the lawyer who represents you. The initial yod is quiet in the verbs but resurfaces in the noun יִצּוּג. Every word is verified against pealim, niqqud included.
The same root runs through the verb system. Each binyan is a voice: it bends the one idea into active, passive, causative or reciprocal. Tap to hear.
The two verbs split the meaning cleanly: pi'el לְיַצֵּג is to represent (a lawyer, a delegate), while hif'il לְהַצִּיג is to present or display (a play, a slide, an argument). In the hif'il the first root letter yod assimilates and vanishes from the spelling, a normal trick of initial-yod roots. The passive and reflexive binyanim are not lexicalized here.
Same root letters, different noun moulds (mishkalim). The mould tells you the function: who does it, the act of doing it, or the thing produced.
Strip the prefixes, suffixes and vowels. Three consonants remain, and the meaning lives there.