Homotopy between loop and composite with constant loop

Existential version
Suppose $$x_0$$ is a point in a topological space $$X$$ and $$f$$ is a loop based at $$x_0$$, i.e., $$f$$ is a continuous map from $$[0,1]$$ to $$X$$ such that $$f(0) = f(1) = x_0$$. Suppose $$e$$ is the constant loop based at $$x_0$$, i.e., the loop that stays at $$x_0$$ throughout.

Denote by $$*$$ the composition of loops by concatenation. Then, $$f$$ is homotopic to the loops $$e * f$$ and $$f * e$$.

This statement is essentially the identity element part of the proof that the fact about::fundamental group of a based topological space is indeed a group.

Constructive/explicit version
For a loop $$f$$ based at $$x_0$$, the loop $$e * f$$ is given by:

$$\! (e * f)(t) = \lbrace\begin{array}{rl} x_0, & 0 \le t \le 1/2\\ f(2t - 1), & 1/2 < t \le 1 \\\end{array}$$

The homotopy between $$f$$ and $$e * f$$ is given by:

$$\! F_\ell(t,s) = \lbrace\begin{array}{rl} x_0, & 0 \le t \le s/2 \\ f\left(\frac{2t - s}{2 - s}\right), & s/2 < t \le 1 \\\end{array}$$

The loop $$f * e$$ is given by:

$$\! (f * e)(t) = \lbrace\begin{array}{rl} f(2t), & 0 \le t \le 1/2\\ x_0, & 1/2 < t \le 1 \\\end{array}$$

The homotopy between $$f$$ and $$f * e$$ is given by:

$$\! F_r(t,s) = \lbrace\begin{array}{rl} f\left(\frac{2t}{2 - s}\right), & 0 \le t \le (2 - s)/2 \\ x_0, & (2 - s)/2 < t \le 1\\\end{array} $$

Graphical version
Here is a pictorial description of the homotopy between $$f$$ and $$e * f$$:



Here is a pictorial description of the homotopy between $$f$$ and $$f * e$$: