Operations on functions
Robert P. André · 2025
In this section we define the composition , https://www.w3.org/1998/Math/MathML" display="inline"> g ∘ f https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/281a012d-36b4-4741-9e04-b0d87b5813f6/content/math10_1.tif "/> , of two functions https://www.w3.org/1998/Math/MathML" display="inline"> f : A → B https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/281a012d-36b4-4741-9e04-b0d87b5813f6/content/math10_2.tif "/> and https://www.w3.org/1998/Math/MathML" display="inline"> g : B → C https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/281a012d-36b4-4741-9e04-b0d87b5813f6/content/math10_3.tif "/> . We view “composition of functions” as an operation “ https://www.w3.org/1998/Math/MathML" display="inline"> ∘ https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/281a012d-36b4-4741-9e04-b0d87b5813f6/content/math10_4.tif "/> ” on two functions f and g. From this perspective we then discuss the main properties of composition of functions (such as non-commutativity and associativity). It is in this particular context that we describe the identity function and the inverse of a function. We also define the concept of “invertible function”.