Why do these two functions have different graph?

Question

Given g(x+1)=x^3, f(x) = (x-1)^3. Why do they have different graphs? In other words, for function g, when x=0, its result is 0, instead for function f, its value is -1, and when x=1,g(x+1)'s result is 1 while f(x)'s result is 0, and so forth, which obviously they will have different graphs. However, In my mind, the function g(x+1) is just a variant of f(x). So, I wonder whether the graph of g(x+1) should be the same as f(x) or not.

Answer 1

The OP presents the function $g$ in an unusual form by giving it the argument $x+1$, whereas the argument $x$ is standard in mathematics. Still, in my opinion this choice is quite acceptable. Unfortunately the OP then gets confused on how to plot the graphs of $f$ and $g$.

Let us choose some $x$ values and evaluate $f$ and $g$:

$$x=0; \space g(1)=0; \space f(0) = -1$$ $$x=1; \space g(2) = 1; \space f(1) = 0$$ $$x = 2; \space g(3) = 8; \space f(2)=1$$ $$x = 3; \space g(4) = 27; \space f(3)=8$$

We verify that $g(1)=f(1) = 0$, $g(2) = f(2)=1$, $g(3) = f(3)=8$. Therefore the functions are the same and so are there graphs.