Calculus How To

Family of Functions

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A family of functions all share common characteristics. Sorting functions into families is much like an animal taxonomy. In an animal taxonomy, you sort things into a tree like structure, based on characteristics like feathers, legs, or bones. Sorting functions into families works much the same way, except they are sorted by mathematicial characteristic like shape, formula and symbols.

For example, linear functions all produce a line when graphed, radical functions contain a radical sign (√), and power functions have the form f(x) = xp.

Families of Functions and Their Graphs

The following graph shows you the shape of some of the parent functions for function families. The parent function is the basic function that all the “children” are based off of:

a family of functions parents

Parent functions: linear function (red), quadratic function (blue), cubic function (green), absolute value function (purple).

Family of Functions and Their Subsets

A family of functions isn’t a simple set. Some functions fall into more than one family, while other functions are subsets of other functions. For example, the constant function is a family of straight-line functions. But they also belong to the family of polynomial functions, a large family, with many subdivisions.

Polynomial functions are a family of several functions, which all share a series of monomials in common (i.e. one or more terms that are added or subtracted). The polynomials include linear functions, cubic functions and quadratic functions.

polynomial function constant term

A family of second degree polynomials.

There are hundreds of function families. For a full list, see: Types of Functions.

Stephanie Glen. "Family of Functions" From Calculus for the rest of us!

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