function

• is a mapping between 2 sets where every element of a first set (domain) maps exactly to one element in the second set (co-domain)

• 𝑓: ℝ → ℝ
• 𝑓(𝑥) = 𝑥²

• 𝑓: ℝ\{0} → ℝ
• 𝑓(𝑥) = 1/𝑥

• 𝑓:ℝ²→ℝ
• 𝑓(𝑥₁, 𝑥₂) = 𝑥₁ + 2𝑥₂
• 𝑓(𝑥₁, 𝑥₂) = [1, 2][𝑥, 𝑥]^𝑇

domain

• the set into which all of the input of the function is constrained to fall

• domain of 𝑓 is ℝ

• domain of 𝑓 is ℝ\{0}

• domain of 𝑓 is ℝ²

codomain

• the set into which all of the output of the function is constrained to fall

• codomain of 𝑓 is ℝ

• codomain of 𝑓 is ℝ

• codomain of 𝑓 is ℝ

preimage

• preimage of a function is the set of all input values (i.e. subset of domain)
• preimage of an element 𝑦 in the domain 𝑌 is defined to be {𝑥 | 𝑓(𝑥)=𝑦}

• preimage of function 𝑓 is ℝ
• preimage of element 4 is 2

• preimage of function 𝑓 is ℝ\{0}
• preimage of element 1/2 is 2

• preimage of function 𝑓 is ℝ²
• preimage of element [4] are the set of elements that satisfy [2,1] + [-2𝑎, 𝑎] for ∀𝑎∊ℝ
• preimage of element [4] is the element [2, 1]

image

• image of a function is the set of all output values (i.e. subset of codomain)
• image of an element 𝑥 in the domain 𝑋 is defined to be {𝑦|𝑓(𝑥)=𝑦}

• image of function 𝑓 is non-negative reals
• image of element 2 is 4

• image of function 𝑓 is ℝ\{0}
• image of element 2 is 1/2

• image of function 𝑓 is ℝ
• image of element [2, 1] is the element [4]
• image of element [0, 2] is the element [4]

range

• either image or codomain