To be a mediator, to be a complete Savior, He had to be appointed by the Father to all three (Prophet, Priest, King) and equipped by the Spirit for all three offices.
The truth is that the idea of humanness already encompasses within itself this threefold dignity and activity. Human beings have a head to know, a heart to give themselves, a hand to govern and to lead; corresponding, they were in the beginning equipped by God with knowledge and understanding, with righteousness and holiness, with dominion and glory (blessedness). The sin that corrupted human beings infected all their capacities and consisted not only in ignorance, folly, error, lies, blindness, darkness but also in unrighteousness, guilt, moral degradation, and further in misery, death, and ruin. Therefore, Christ, both as the Son and as the image of God, for Himself and also as our Mediator and Savior, had to bear all three offices.
He had to be a prophet to know and disclose the truth of God. He had to be a priest, to devote Himself to God and, in our place, to offer Himself up to God. He had to be a king, to govern and protect us according to God’s will. To teach, to reconcile, and to lead; to instruct, to acquire, and to apply salvation; wisdom, righteousness, and redemption; truth, love, and power – all three are essential to the completeness of our salvation.
In Christ’s God-to-Humanity relation, He is a prophet; In His Humanity-to-God relation He is a priest; in His headship over all humanity He is a king … Scripture, consistently and simultaneously attributing all three offices to him, describes Him as our chief prophet, our only [high] priest, and our eternal king. Though a King, He rules not by the sword, but by His Word and Spirit. He is a Prophet, but His Word is power and [really] happens. He is a Priest, but lives by dying, conquers by suffering, and is all-powerful by His love. He is always all these things in conjunction, never the one without the other: mighty in speech and actions as a king and full of grace and truth in his royal rule.
Source: Bavinck, Reformed Dogmatics, Volume Three: Sin and Salvation in Christ, Christ’s Humiliation, Christ’s Threefold Office p. 367-36