Examples of functional importance in the following topics:

 Functional groups are groups of molecules attached to organic molecules and give them specific identities or functions.
 A functional group can participate in specific chemical reactions.
 Some of the important functional groups in biological molecules include: hydroxyl, methyl, carbonyl, carboxyl, amino, phosphate, and sulfhydryl groups.
 These groups play an important role in the formation of molecules like DNA, proteins, carbohydrates, and lipids.
 Hydrogen bonds between functional groups (within the same molecule or between different molecules) are important to the function of many macromolecules and help them to fold properly and maintain the appropriate shape needed to function correctly.

 The derivative of the exponential function is equal to the value of the function.
 The importance of the exponential function in mathematics and the sciences stems mainly from properties of its derivative.
 Functions of the form $ce^x$ for constant $c$ are the only functions with this property.
 The rate of increase of the function at $x$ is equal to the value of the function at $x$.
 Graph of the exponential function illustrating that its derivative is equal to the value of the function.

 Functional groups are atoms or small groups of atoms (two to four) that exhibit a characteristic reactivity when treated with certain reagents.
 A particular functional group will almost always display its characteristic chemical behavior when it is present in a compound.
 Because of their importance in understanding organic chemistry, functional groups have characteristic names that often carry over in the naming of individual compounds incorporating specific groups.
 In the following table the atoms of each functional group are colored red and the characteristic IUPAC nomenclature suffix that denotes some (but not all) functional groups is also colored.

 Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse.
 This is not true of the function $f(x)=x^2$.
 Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse.
 The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
 No, the function has no defined value for $x=0$.

 Functional groups also play an important part in organic compound nomenclature; combining the names of the functional groups with the names of the parent alkanes provides a way to distinguish compounds.
 Functionalization refers to the addition of functional groups to a compound by chemical synthesis.
 In materials science, functionalization is employed to achieve desired surface properties; functional groups can also be used to covalently link functional molecules to the surfaces of chemical devices.
 It is important to be able to recognize the functional groups and the physical and chemical properties that they afford compounds.
 Alcohols are a common functional group (OH).

 Hyperbolic function is an analog of the ordinary trigonometric function, also called circular function.
 The basic hyperbolic functions are the hyperbolic sine "$\sinh$," and the hyperbolic cosine "$\cosh$," from which are derived the hyperbolic tangent "$\tanh$," and so on, corresponding to the derived functions.
 Hyperbolic functions occur in the solutions of some important linear differential equations, for example the equation defining a catenary, of some cubic equations, and of Laplace's equation in Cartesian coordinates.
 The latter is important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.
 When considered defined by a complex variable, the hyperbolic functions are rational functions of exponentials, and are hence meromorphic.

 That is, the function divides the input by two.
 The function machine allows us to alter expressions.
 This is an important fact about functions that cannot be stressed enough: every possible input to the function must have one and only one output.
 All functions are relations, but not all relations are functions.
 Connect the notation of functions to the notation of equations generally and give the criteria for being a function

 Functional composition allows for the application of one function to another; this step can be undone by using functional decomposition.
 The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions.
 The resulting function is known as a composite function.
 It is also important to understand the order of operations in evaluating a composite function.
 Practice function composition by applying the rules of one function to the results of another function

 Each trigonometric function has an inverse function that can be graphed.
 In order to use inverse trigonometric functions, we need to understand
that an inverse trigonometric function “undoes” what the original
trigonometric function “does,” as is the case with any other function
and its inverse.
 However, the sine, cosine, and tangent functions are not
onetoone functions.
 As with other
functions that are not onetoone, we will need to restrict the domain of each function to yield a new function that is onetoone.
 These conventional choices for the restricted domain are somewhat
arbitrary, but they have important, helpful characteristics.

 A particular functional group will almost always display its characteristic chemical behavior when it is present in a compound.
 Because of this, the discussion of organic reactions is often organized according to functional groups.
 The following table summarizes the general chemical behavior of the common functional groups.
 For reference, the alkanes provide a background of behavior in the absence of more localized functional groups.
 Consequently, apparent similarities may fail to reflect important differences.