Function operations and composition of functions
PRESS START
section 01
Function operations
Let π and π be any two functions. You can add, subtract, multiply π(π) and π(π) to form a new function.
The domain of new function consist of the π -values that are in the domains of both π(π) and π(π).
function operations and composition
function operations and composition
Example n.1
Find (π+π)(π), (πβπ)(π),(πβπ)(π) for each π(π) and π(π). a.π(π)=π^π+ππβπ π(π)=πβπ
function operations and composition
Example n.1
Find (π+π)(π), (πβπ)(π),(πβπ)(π) for each π(π) and π(π). a.π(π)=π^π+ππβπ π(π)=πβπ
(π+π)(π)=(π^π+ππβπ)+(πβπ)
(π+π)(π)=π^π+ππβπ
function operations and composition
Example n.1
Find (π+π)(π), (πβπ)(π),(πβπ)(π) for each π(π) and π(π). a.π(π)=π^π+ππβπ π(π)=πβπ
(πβπ)(π)=(π^π+ππβπ)β(πβπ) (πβπ)(π)=π^π+ππβπβπ+π
(πβπ)(π)=π^π+π+π
function operations and composition
Example n.1
Find (π+π)(π), (πβπ)(π),(πβπ)(π) for each π(π) and π(π). a.π(π)=π^π+ππβπ π(π)=πβπ
(πβπ)(π)=(π^π+ππβπ)β(πβπ)
(πβπ)(π)=π^πβππ^πβπππ+π
function operations and composition
Example n.2
Find (π+π)(π), (πβπ)(π),(πβπ)(π) for each π(π) and π(π). b. π(π)=π^πβππ π(π)=π+π
function operations and composition
Example n.2
Find (π+π)(π), (πβπ)(π),(πβπ)(π) for each π(π) and π(π). b. π(π)=π^πβππ π(π)=π+π
(π+π)(π)=(π^πβππ) +(π+π) (π+π)(π)=π^π+πβππ
function operations and composition
Example n.2
Find (π+π)(π), (πβπ)(π),(πβπ)(π) for each π(π) and π(π). b. π(π)=π^πβππ π(π)=π+π
(πβπ)(π)=(π^πβππ)β(π+π) (πβπ)(π)=π^πβππβπβπ (πβπ)(π)=π^πβπβππ
function operations and composition
Example n.2
Find (π+π)(π), (πβπ)(π),(πβπ)(π) for each π(π) and π(π). b. π(π)=π^πβππ π(π)=π+π
(πβπ)(π)=(π^πβππ)β(π+π) (πβπ)(π)=π^π+ππ^πβπππβπππ
composition of functions
The composition of function π with function π is defined by (πβπ)(π)=π(π(π))
The domain of the composite function πβπ is the set of all such that:
1. π is in the domain of π and
2. π(π) is in the domain of π.
composition of functions
f f(g(x))
g g(x)
g(x) must be in the domain of f
composition of functions
Example n.1
Find each composite function. a.π(π)=ππβπ π(π)=π+π
(πβπ)(π)=?
composition of functions
Example n.1
Find each composite function. a.π(π)=ππβπ π(π)=π+π
(πβπ)(π)=?
(πβπ)(π)=π(π(π)) π(π(π))=π(π(π))βπ
π(π(π))=π(π+π)βπ
π(π(π))=ππ+πβπ
π(π(π))=ππβπ
composition of functions
Example n.2
Find each composite function. b.π(π)=πβπ π(π)=π^π+π
(πβπ)(π)=?
composition of functions
Example n.2
Find each composite function. b.π(π)=πβπ π(π)=π^π+π
(πβπ)(π)=? (πβπ)(π)=π(π(π)) π(π(π))=(π(π))^π+π
π(π(π))=(πβπ)^π+π
π(π(π))=π^πβππ+π+π
π(π(π))=π^πβππ+ππ
composition of functions
Example n.3
Find each composite function. c.π(π)=π/(πβπ) π(π)=π/π (πβπ)(π)=?
(πβπ)(π)=π(π(π))
π(π(π))=π/(π(π)βπ) π/πβ π πβ π
π(π(π))=π/(π/πβπ) π/πβπβ π πβ π/π
π(π(π))=ππ/(πβππ)
composition of functions
Example n.4
Find each composite function. d. π(π)=π/π π(π)=π/π (πβπ)(π)=?
(πβπ)(π)=π(π(π)) π(π(π))=π/π(π) π/πβ π πβ π
π(π(π))=π/(π/π) π(π(π))=π/π
composition of functions
Example n.5
Find and then evaluate each composite function.e.π(π)=βπ π(π)=πβπ (πβπ)(π)=?
composition of functions
Example n.5
Find and then evaluate each composite function.e.π(π)=βπ π(π)=πβπ (πβπ)(π)=?
(πβπ)(π)=π(π(π))
π(π(π))=β(π(π) ) π(π(π))=β(πβπ) π(π(π))=β(πβπ)π(π(π))=βπ
π(π(π))=π
composition of functions
Example n.5
Find and then evaluate each composite function.f.π(π)=ππβπ π(π)=(π+π)/π (πβπ)(π)=?
(πβπ)(π)=π(π(π)) π(π(π))=(π(π)+π)/π
π(π(π))=((ππβπ)+π)/π
π(π(π))=(ππ+π)/π=(π(ππ+π))/π
π(π(π))=ππ+π
π(π(π))=πβπ+π
π(π(π))=π
GAME OVER
thank you!
FUNCTION OPERATION AND COMPOSITION
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Transcript
Function operations and composition of functions
PRESS START
section 01
Function operations
Let π and π be any two functions. You can add, subtract, multiply π(π) and π(π) to form a new function. The domain of new function consist of the π -values that are in the domains of both π(π) and π(π).
function operations and composition
function operations and composition
Example n.1
Find (π+π)(π), (πβπ)(π),(πβπ)(π) for each π(π) and π(π). a.π(π)=π^π+ππβπ π(π)=πβπ
function operations and composition
Example n.1
Find (π+π)(π), (πβπ)(π),(πβπ)(π) for each π(π) and π(π). a.π(π)=π^π+ππβπ π(π)=πβπ (π+π)(π)=(π^π+ππβπ)+(πβπ) (π+π)(π)=π^π+ππβπ
function operations and composition
Example n.1
Find (π+π)(π), (πβπ)(π),(πβπ)(π) for each π(π) and π(π). a.π(π)=π^π+ππβπ π(π)=πβπ (πβπ)(π)=(π^π+ππβπ)β(πβπ) (πβπ)(π)=π^π+ππβπβπ+π (πβπ)(π)=π^π+π+π
function operations and composition
Example n.1
Find (π+π)(π), (πβπ)(π),(πβπ)(π) for each π(π) and π(π). a.π(π)=π^π+ππβπ π(π)=πβπ (πβπ)(π)=(π^π+ππβπ)β(πβπ) (πβπ)(π)=π^πβππ^πβπππ+π
function operations and composition
Example n.2
Find (π+π)(π), (πβπ)(π),(πβπ)(π) for each π(π) and π(π). b. π(π)=π^πβππ π(π)=π+π
function operations and composition
Example n.2
Find (π+π)(π), (πβπ)(π),(πβπ)(π) for each π(π) and π(π). b. π(π)=π^πβππ π(π)=π+π (π+π)(π)=(π^πβππ) +(π+π) (π+π)(π)=π^π+πβππ
function operations and composition
Example n.2
Find (π+π)(π), (πβπ)(π),(πβπ)(π) for each π(π) and π(π). b. π(π)=π^πβππ π(π)=π+π (πβπ)(π)=(π^πβππ)β(π+π) (πβπ)(π)=π^πβππβπβπ (πβπ)(π)=π^πβπβππ
function operations and composition
Example n.2
Find (π+π)(π), (πβπ)(π),(πβπ)(π) for each π(π) and π(π). b. π(π)=π^πβππ π(π)=π+π (πβπ)(π)=(π^πβππ)β(π+π) (πβπ)(π)=π^π+ππ^πβπππβπππ
composition of functions
The composition of function π with function π is defined by (πβπ)(π)=π(π(π)) The domain of the composite function πβπ is the set of all such that: 1. π is in the domain of π and 2. π(π) is in the domain of π.
composition of functions
f f(g(x))
g g(x)
g(x) must be in the domain of f
composition of functions
Example n.1
Find each composite function. a.π(π)=ππβπ π(π)=π+π (πβπ)(π)=?
composition of functions
Example n.1
Find each composite function. a.π(π)=ππβπ π(π)=π+π (πβπ)(π)=? (πβπ)(π)=π(π(π)) π(π(π))=π(π(π))βπ π(π(π))=π(π+π)βπ π(π(π))=ππ+πβπ π(π(π))=ππβπ
composition of functions
Example n.2
Find each composite function. b.π(π)=πβπ π(π)=π^π+π (πβπ)(π)=?
composition of functions
Example n.2
Find each composite function. b.π(π)=πβπ π(π)=π^π+π (πβπ)(π)=? (πβπ)(π)=π(π(π)) π(π(π))=(π(π))^π+π π(π(π))=(πβπ)^π+π π(π(π))=π^πβππ+π+π π(π(π))=π^πβππ+ππ
composition of functions
Example n.3
Find each composite function. c.π(π)=π/(πβπ) π(π)=π/π (πβπ)(π)=? (πβπ)(π)=π(π(π)) π(π(π))=π/(π(π)βπ) π/πβ π πβ π π(π(π))=π/(π/πβπ) π/πβπβ π πβ π/π π(π(π))=ππ/(πβππ)
composition of functions
Example n.4
Find each composite function. d. π(π)=π/π π(π)=π/π (πβπ)(π)=? (πβπ)(π)=π(π(π)) π(π(π))=π/π(π) π/πβ π πβ π π(π(π))=π/(π/π) π(π(π))=π/π
composition of functions
Example n.5
Find and then evaluate each composite function.e.π(π)=βπ π(π)=πβπ (πβπ)(π)=?
composition of functions
Example n.5
Find and then evaluate each composite function.e.π(π)=βπ π(π)=πβπ (πβπ)(π)=? (πβπ)(π)=π(π(π)) π(π(π))=β(π(π) ) π(π(π))=β(πβπ) π(π(π))=β(πβπ)π(π(π))=βπ π(π(π))=π
composition of functions
Example n.5
Find and then evaluate each composite function.f.π(π)=ππβπ π(π)=(π+π)/π (πβπ)(π)=? (πβπ)(π)=π(π(π)) π(π(π))=(π(π)+π)/π π(π(π))=((ππβπ)+π)/π π(π(π))=(ππ+π)/π=(π(ππ+π))/π π(π(π))=ππ+π π(π(π))=πβπ+π π(π(π))=π
GAME OVER
thank you!