
An implicit definition of a function is one which does not present an explicit formula for its values but rather defines it by giving conditions that it satisfies. Thus its values must be inferred as consequences of the definition, so it is defined "by implication".
An example is: define $y(x)$ by ${x}^{2}+{y}^{2}=1$ and $y0$ .
Defining a function as the inverse is another example of an implicit definition.
Notice that you can produce a formula for $y(x)$ in the example here; that formula represents an explicit definition of this same function.
Exercises:
1.9 How much of this was familiar to you?
1.10 The statement $\mathrm{cos}x$ is $\mathrm{sin}(\frac{\pi}{2}x)$ implies what about $\mathrm{arccos}y$ and $\mathrm{arcsin}y$ ? Solution
1.11 Invent a problem to go here.
