Show that any continuous map $f: \mathbb{R}^n \rightarrow X$ is homotopic to the constant map

Question

$\newcommand {\R}{\mathbb{R}}$ I want to show that any continuous map $f: \R^n \rightarrow X$ is homotopic to the constant map.

So far this is what I have:

My homotopy must look something like that:

$H(x,0) := f(x)$

$H(x,1) :=\gamma_0$

where $\gamma_0$ is in the same path connected component as the image of $\R^n$.

My question now is: could someone help me to formally write down the homotopy? Thanks!

Edit: I took out my intuition for $H$ that I had described earlier as it was wrong and I do not wish to confuse future readers.

Answer 1

$f(t, x) = f((1-t) x)$ should do the trick.