In Genesis 2:7, we read "man became a living soul," and in Job 14:22, we read "his soul within him." Is the soul a part of man, or is the whole of man a living soul?

What is the soul? It is the life-giving quality, the energizing principle, in union with the organism, or body, forms the living, intelligent being: BODY + BREATH OF LIFE = SOUL.  (Genesis 1:20,30; Genesis 2:7) 

The word soul in this connection is from the Hebrew which means literally “a breathing creature.” Man does not possess a soul, but he is a soul. The entire being is the soul. Job’s expression must be understood in the poetic sense and signifies that man would suffer and mourn from his inmost being, as in, “from the bottom of his soul.” 

Additional Resource:
Christian Questions Podcast Episode #1227: “Where Do the Human Soul and Spirit Go When We Die? (Part I)”
Defining what the human soul is and is not, and its destiny at death
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CQ Rewind Show Notes