Counterexample : A counterexample to a statement of the form 'all objects of type x have property y' is an object of type x that does not have property y. A counterexample to a statement shows that the statement is not true in general. For example the number 3 is a counterexample to the statement that all numbers are even.

Function : A function is a mapping in which every element in one set is mapped to exactly one element of a second set. Most often the mapping is described using a rule. For example the function $f(x)=x+1$ maps 2 to 3 and -1 to 0.

Root : A root is a solution to an equation of the $f(x)=0$ where $f$ is a function or expression. For example the function $f(x)=x-1$ has the root $x=1.$

Function : A function is a mapping in which every element in one set is mapped to exactly one element of a second set. Most often the mapping is described using a rule. For example the function $f(x)=x+1$ maps 2 to 3 and -1 to 0.

Graph : The graph of a function is the set of points $\{(x,f(x))\}$ that can be blotted in a coordinate system.

Function : A function is a mapping in which every element in one set is mapped to exactly one element of a second set. Most often the mapping is described using a rule. For example the function $f(x)=x+1$ maps 2 to 3 and -1 to 0.

Monotonically decreasing : A function $f$ is called monotonically decreasing if we have $f(x)\geq f(y)$ for every $x\lt y.$ For example the function $f(x)=-x$ is monotonically decreasing and so is the function $f(x)=e^{-x}.$ A differentiable function $f$ is monotonically decreasing if and only if $f^{\prime}\leq 0.$

Monotonically increasing : A function $f$ is called monotonically increasing if we have $f(x)\leq f(y)$ for every $x\lt y.$ For example the function $f(x)=x$ is monotonically increasing and so is the function $f(x)=x^3.$ A differentiable function $f$ is monotonically increasing if and only if $f^{\prime}\geq 0.$

Negative : A number is negative if it is $\lt 0.$

Positive : Positive means $\geq 0.$