During antiquity, particular examples of relationships and problems between two things were studied and solved, but no generalized ideas of these problems were formed. There was no abstract idea of a variable to help do this, therefore quantities were described verbally or with a graph instead of a formula.
The Babylonians in 2000 B.C. had tables of reciprocals, squares, square roots, cubes, and cube roots, and many other things. The Babylonian astronomers kept their calculations on tables too.
The Greek concept of function developed around many things. They studied the laws of acoustics and developed a table of chords. They did this by studying the interdependence of the length and pitches of notes emitted my plucked strings of the same type under equal tension. Astronomers developed sine tables equivalent to ours today. They studied geometry and curves. They calculated areas, volumes, lengths, and centers of gravity. Roots of polynomials were solved using conic sections.
Problems involving change and variation were studied. They looked at motion, continuity, and infinity. All problems were specific and explained verbally in a table, graph, or by example. It seems they looked at problems that were functions, but they lacked the word "function," and without symbolism, expressing their ideas as analytical expressions or formulas wasn't possible.