Final answer:
The domain and range of a function are affected differently by transformations such as shifts, stretches, compressions, and reflections.
Step-by-step explanation:
The question asks how domain and range change with transformations in a mathematical context. Transformations refer to changes in the graphical representation of functions, including shifting, stretching, compressing, and reflecting. When a function is subjected to a transformation, its domain and range can be affected differently depending on the type of transformation applied.
For example, a horizontal shift does not affect the range but does change the domain. If a function f(x) is shifted horizontally by k units, the new function g(x) = f(x - k) will have the same range as f(x) but its domain will be shifted by k units.
A vertical shift, on the other hand, affects the range but not the domain. If f(x) is shifted vertically by k units, the new function g(x) = f(x) + k will have its range shifted by k units, while maintaining the original domain.
Stretching or compressing a function vertically multiplies the range by a factor, while stretching or compressing horizontally divides the domain by a factor. Reflecting a function over the x-axis or y-axis changes the sign of all the y-values (range) or x-values (domain), respectively.
Understanding how these transformations alter domain and range is important in graphically representing functions and in solving equations. With practice, anticipating the effects of transformations on domain and range becomes an intuitive part of problem-solving.