What is the Algebraic proof of the horizontal shift rule

Question

I know the Geometric proof just by plotting and looking at the graph but what is the Algebraic proof of the Horizontal shift rule? Do not provide examples or restate the rules or say that the Algebraic proof is f(x+c) goes left and f(x-c) goes right. I know that already. What is the proof for all type of functions?

Answer 1

Try letting g(x) = f(x+c)

Notice that g(x-c) = f(x)

This is saying that the value of f at x is the same as the value of g at x-c. In other words, c to the left of x -- a horizontal shift of c to the left.

Vice versa for subtracting c:

let g(x) = f(x-c), then g(x+c) = f(x)

The value of f at x is the same as the value of g at x+c. In other words, c to the right of x -- a horizontal shift of c to the right.