A student was given the function f(x) = x^2 and had to write the result of translating the function 1 unit to the left. Is g(x) = (x +1)^2 the right answer.
The graph of f(x) = x^2 (in black) and g(x) = (x +1)^2 (in red) is:
It is seen that g(x) is shifted to the left by 1 unit. The function g(x) satisfies the given criterion. g(x') = (x' + 1)^2 has the same value as that of f(x) for any value of x where x' = x - 1.
This shows that g(x) = (x +1)^2 is the right answer.