Finding the Inverse Image of a Mapping

List the elements of the complete inverse image $\theta^{-1}(5)$ under the mapping $\theta(x) = |x|$.

In this problem, we explore the concept of the inverse image of a set under a given mapping. Specifically, we look at the mapping theta of x which equals the absolute value of x and find the elements that map to the number 5. This involves understanding how the absolute value function operates and considering both the positive and negative inputs that can yield the same output when passed through the absolute value function.

This type of problem is fundamental in understanding mappings and functions because it challenges you to think about how functions can take an input and relate it to an output, and more specifically, how you can determine the set of all possible inputs for a given output. It helps solidify the concept of inverse operations and functions, a crucial aspect of function theory. Solving such a problem requires a good grasp of the properties of absolute values and a systematic approach to identifying all elements of the domain of the inverse image.