Function composition

Function composition combines two functions by feeding the output of one into the other. The resulting function applies the inner function first and the outer function second.

Composite function: A function formed by applying one function to the output of another. It is the function produced by the composition operation.
Function: A rule assigning exactly one output to each input in its domain. Composition builds a new function from existing functions.
Affine transformation: A map between affine spaces formed by a linear transformation followed by a translation. Composing affine transformations combines geometric actions into one map.
Pointwise multiplication of functions: The function formed by multiplying the values of two functions at each input. Unlike composition, pointwise multiplication combines outputs evaluated at the same input.
Domain of a function: The set of inputs for which a function is defined. An input must lie in the inner function’s domain, and its output must suit the outer function.
Function notation: A symbolic way to denote a function and its value at a specified input. The expression f(g(x)) shows that g acts before f.
Polynomial function: A function defined by evaluating a polynomial expression at an input. Substituting one polynomial into another is function composition.
Pointwise addition of functions: The function formed by adding the values of two functions at each input. Addition combines paired outputs rather than feeding one function’s output into another.
Codomain: The specified target set in which a function’s outputs are required to lie. The inner function’s codomain helps determine whether its outputs can serve as outer-function inputs.
Image (mathematics): The set of outputs a function attains on a specified set of inputs. The inner function’s image must fit within the outer function’s domain.

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