I've stumbled upon a question which asks me to prove that $f_0 := g*c$ and $f_1 := c*g$ are homotopic. More specifically it wants me to give a 'picture in $I\times I$, a picture in $X$ and an explicit homotopy showing that $f_0$ and $f_1$ are homotopic'.
$g$ is defined to be a loop $g:I\to X$ starting at $x$ and $c:I\to X$ which is a constant loop starting at $x$.
I'm unsure of what they mean by the explicit homotopy and the picture in $I\times I$. My guess would be that the picture in $I\times I$ for $g*c$ is a movement upwards between $0$ and $1/2$ and then constant from $1/2$ to $1$ but I'm not sure at all. Surely the picture in $X$ is just the graph of $g$, a loop?
Any assistance would be greatly appreciated.
Thanks in advance
Let $$\phi(x)=\begin{cases}g(x)&\text{if $0\le x\le 1$}\\g(0)&\text{otherwise}\end{cases}$$ Then consider $f(s,t)=\phi(2s-t)$.