Vipul: Created page with "==Definition== Suppose $f,g$ are two functions. The '''composite function''' $f \circ g$ is defined as the function: $f \circ g = x \mapsto f(g(x..."$

2011-08-26T10:59:00Z

Created page with "==Definition== Suppose <math>f,g</math> are two functions. The '''composite function''' <math>f \circ g</math> is defined as the function: <math>f \circ g = x \mapsto f(g(x..."

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==Definition==

Suppose <math>f,g</math> are two [[function]]s. The '''composite function''' <math>f \circ g</math> is defined as the function:

<math>f \circ g = x \mapsto f(g(x))</math>

Note that the function written at the right end of the composition is the function performed ''first'', and the function written at the left end of the composition is the function performed ''next''. We say that composition of functions is ''done right-to-left''.

==Relation with various operations==

Below, we discuss how a particular operation done for functions can be done for a composite of two functions:

{| class="sortable" border="1"
! Operation !! Verbal description !! How it's done
|-
| Graph <math>f \circ g</math> || We are given the graphs of <math>f</math> and <math>g</math> (without necessarily having algebraic, numerical, or verbal descriptions of the functions) and we need a geometric method to sketch the graph of <math>f \circ g</math> || [[graphing the composite of two functions]]
|-
| Obtain explicit expression for <math>f \circ g</math> || We are given explicit algebraic expressions for <math>f</math> and <math>g</math> and need an explicit algebraic expression for <math>f \circ g</math>. || simple case: [[finding the composite of two functions by plugging in expressions]]<br>case of piecewise functions: [[finding the composite of two piecewise functions]]
|-
| Find limit of <math>f \circ g</math> at a point || We have techniques for finding limits for both functions, we need a technique for finding the limit of the composite. || [[composition theorem for continuous functions]]
|-
| Differentiate <math>f \circ g</math>. || We have expressions for the [[derivative]]s <math>f'</math> and <math>g'</math>, we need an expression for <math>(f \circ g)'</math>. || [[chain rule for derivatives]]: <math>(f \circ g)' = (f' \circ g) \cdot g'</math>.
|-
| Integrate <math>f \circ g</math>. || We want to integrate <math>f \circ g</math> in terms of integration of simpler functions. || We can try [[integration by u-substitution]] or [[integration by parts]].
|}

Composite of two functions - Revision history

Vipul: Created page with "==Definition== Suppose f,g are two functions. The '''composite function''' f \circ g is defined as the function: f \circ g = x \mapsto f(g(x..."

Vipul: Created page with "==Definition== Suppose $f,g$ are two functions. The '''composite function''' $f \circ g$ is defined as the function: $f \circ g = x \mapsto f(g(x..."$