1.1.2 A subtlety
Two functions are said to be equal if they have the same set of inputs and each input is mapped to the same output. That is, the functions and are equal if their domains are the same and for every element in the common domain, . Note that we did not say the two functions are equal if their domains and their images are equal. Think about why this second definition is not strong enough. Can you think of two functions with the same domain and same image that are not equal?
In what ways are the two functions displayed above not equal?
Some questions to ponder. Think about all the knowledge you have of the two functions and . Is there any way to glean that information from the animation above? Specifically is it possible to look at the animation above and see that: 1. Both functions fail to be injective. 2. Both functions have an absolute minimum at . 3. is differentiable everywhere, but fails to be differentiable at . 4. The derivative of is a variable linear function while the derivative of (where it exists) is one of just two different constant functions. 5. is concave up across its domain, but has no concavity anywhere on its domain.