Just getting started with Hatcher, wondering if I get the right idea at the beginning?
Problem 0.2, Page 18:
Construct an explicit deformation retraction of $\mathbb{R}^n – \{0\}$ onto $S^{n−1}$.
Consider
$$f_t: X \to X, t \in I, f_t(x) = (1-t)x + t \frac{x}{|x|}.$$
Hence, $f_0 = \mathbb{I}$, $f_1 = \frac{x}{|x|}$, and $f_t|_{S^{n-1}} = S^{n-1}$ for all $t$.
Best Answer
Yes your solution is correct. What you could do additionally is to argue why $f\mid_{S^{n-1}}=id$ for all $t$. It can be argued as follows: if $x \in S^{n-1}$ then $x = {x \over |x|}$ and therefore $f(x) = (1-t)x + tx = x$.