In the answer(source - Campbell Essential Biology with Physiology, self-quiz question), it's given that "if the change doesn't affect the protein's shape in any way then it's possible to do so". But practically, is it possible to amend a protein without changing its shape?
What the authors are describing here is the idea that function follows from form. Meaning that a proteins 3D structure is, in many ways, responsible for it's actions.
And yes, it is possible to change a residue without changing the proteins shape. For example, changing a residue that resides on the outside of the protein from a positively charged one like Lysine to a positively charged one like Arginine (both residues are very similar) will not have any effect on the structure of the protein, in most cases. Same thing when changing an Alanine to a Valine: they are so similar to one another that it would be difficult to find a case where swapping them would cause the protein to change shape or for its function to be destroyed.
Although the question is vague, there is value in it.
We have to distinguish shape from function. The function of a protein is mostly (but not only) attributed to certain discrete regions, called domains, which have a particular "shape" attributable to the protein's primary structure (i.e. amino acids) which arrange in a particular 3D spatial conformation (this is the protein's "shape", as explained in your book). One may change an amino acid (mutate it) within a domain without it losing its shape and function. Such a change should be conservative (e.g. between similar amino acids), but sometimes even an additional carbon atom (e.g. Gly -> Ala) or carbon permutation (e.g. Leu -> Ile) may change a protein's function. Could you change the protein's shape through a mutation? Certainly. It may or may not lose/affect its function(s). One can even delete entire portions of a protein without it losing its function(s). As John says in the comments, we do this in laboratory (practically) to see the effects of amino acid changes (mutations).
As I can see, in your textbook, you see this, and here is where the value lies in:
Q: Is it possible to change an amino acid within a protein but not affecting that protein's functions?
A: If the change doesn't affect the protein's shape in any way then it's possible to do so.
I do not have access to your textbook, but the answer there is actually false (or incorrect). Why? Because a protein's shape does not dictate its proper function. We may change its "shape" (3D conformation) of a non-functional domain (or a linker, or an non-important loop) and its function(s) can be wholly preserved.
PS I do not want to mislead you, but the answer provided there may be useful for beginners in Biology. A beginner's book may not dwelve into such intricancies, so it leaves its reader associate protein shape with its function. Albeit incorrect, it simplifies things by reduction. Nonetheless, it is hardly acceptable to do so.
PPS Most Biochemistry textbooks cover these aspects through their chapters (e.g. Lehninger's Principles). This study mentions exactly what I said:
A mutation in the amino acid sequence may alter the structure of a protein but it does not necessarily alter its function, although, the mutation at specific sites such as conserved residues can bring about a change in the structure and function of the protein.
Going in more depth, it is even possible to maintain the shape of the protein, but alter its function, for instance, through indirect, allosterical effects.
Although, the conformation of the mutant protein may be highly similar to the conformation of the wildtype, there could be alterations in their topologies at sites distant from the site of mutation.
In this category we also include active sites of enzymes, whereby proper amino acid residues interact with the substrate; their mutation does not change the shape, but can directly perturb/abolish the enzyme's activity (example).