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Summary of aspects of functions: domain, range, domain restrictions. Worked out for all elementary mathematical functions: linear, quadratic, etc.
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Functions&
Graphs
Photo: www.flickr.com
Functions & Graphs 2
Functions
Relations
Functions & Graphs 3
Domain & Range
𝒟= {−3 ,−2,0,1 }
ℛ= {−1,1,2 }
𝒟=ℝ
ℛ=(−∞, 4 12 ]
Domain = Set of independent variablesRange = Set of dependent variables
Functions & Graphs 4
Notation
𝑓 :𝑥⟼2𝑥2+5
𝑦=2 𝑥2+5
𝑦= 𝑓 (𝑥 )
Argument
𝑓 :𝑥⟼𝑎𝑥2+𝑏
Parameters
Functions & Graphs 5
Domain restriction
𝑓 :𝑥⟼1
𝑥−2𝒟=ℝ
𝒟= {𝑥|𝑥∈ℝ∧𝑥≠2 }
is NOT a function on
IS a function on
Functions & Graphs 6
Domain by context𝑣 (𝑡 )=100
𝑡Average velocity
is mathematically a function on
is by context a function on
Photo: Wikipedia.org
Functions & Graphs 7
Linear functions𝑓 :𝑥⟼𝑎𝑥+𝑏𝒟=ℝℛ=ℝ
𝑏
∆ 𝑥
∆ 𝑦
𝑎=∆ 𝑦∆ 𝑥
Straight Line
Functions & Graphs 8
Quadratic functions - I𝑓 :𝑥⟼𝑎𝑥2+𝑏𝑥+𝑐𝒟=ℝℛ=[ 𝑦𝑣 , ∞ )
(𝑥𝑣 , 𝑦𝑣 )=(− 𝑏2𝑎, 𝑓 (− 𝑏
2𝑎 ))
Axis of symmetry
Vertex
𝑎>0
Vertex =
Parabola
Functions & Graphs 9
Quadratic functions - II𝑓 :𝑥⟼𝑎𝑥2+𝑏𝑥+𝑐𝒟=ℝℛ=(−∞ , 𝑦𝑣 ]
(𝑥𝑣 , 𝑦𝑣 )=(− 𝑏2𝑎, 𝑓 (− 𝑏
2𝑎 ))
Axis of symmetry
Vertex
𝑎<0
Vertex =
Parabola
Functions & Graphs
Quadratic functions – Determine vertex
10
𝑦=−12𝑥2+4 𝑥+6
(𝑥𝑣 , 𝑦𝑣 )=(4,14 )⇒ℛ= (−∞, 14 ]
Axis of symmetry
Vertex
Vertex =
Parabola
“Complete squares”
𝑦=−12
[𝑥2−8 𝑥−12 ]
𝑦=−12
[ (𝑥−4 )2−16−12 ]
𝑦=−12
[ (𝑥−4 )2−28 ]
divide by 1st parameter
Include half of 2nd parameter in the square
Compensate for the extra constant
Square term smallest (0), if
𝑦 𝑣=−12 [ (𝑥𝑣−4 )2−28 ]=14
Functions & Graphs 11
Radical & Absolute value functions
𝑓 :𝑥⟼√4 𝑥−3
𝒟=[ 34 , ∞ )
ℛ=[0 , ∞ )
𝑓 :𝑥⟼|𝑥|
𝒟=ℝ
ℛ=[0 , ∞ )
Functions & Graphs 12
Reciprocal & Rational functions
𝑓 :𝑥⟼1𝑥
𝒟= {𝑥|𝑥∈ℝ∧𝑥≠0 }
ℛ= {𝑦|𝑦∈ℝ∧𝑦 ≠0 }
𝑓 :𝑥⟼3 𝑥−42 𝑥+5
𝒟={𝑥|𝑥∈ℝ∧𝑥≠−2 12 }
ℛ={𝑦|𝑦∈ℝ∧ 𝑦 ≠1 12 }
AsymptotesHyperbola
Functions & Graphs 13
Rational functions
𝑓 :𝑥⟼3 𝑥−42 𝑥+5
𝒟={𝑥|𝑥∈ℝ∧𝑥≠−2 12 }
ℛ={𝑦|𝑦∈ℝ∧ 𝑦 ≠1 12 }
AsymptotesFinding the vertical asymptoteFraction is undefined, if denominator = 0
2 𝑥+5=0⇒ 𝑥=−212
Finding the horizontal asymptoteTake a ‘huge’ number for
𝑦=3 ∙10100−4
2 ∙10100+5≈3∙10100
2∙10100=32=1 1
2
Functions & Graphs 14
Many-to-one vs. One-to-one
Functions & Graphs 15
Even vs. Odd
𝑓 (−𝑥 )= 𝑓 (𝑥) 𝑓 (−𝑥 )=− 𝑓 (𝑥 )
For all For all
Functions & Graphs 16
Composite functions𝑓 ∘𝑔 (𝑥 )= 𝑓 (𝑔 (𝑥 ) )
Inner functionOuter function
ℛ𝑔⊂𝐷 𝑓 Is required! If not, must be restricted
ℛ 𝑓𝐷𝑔 𝐷 𝑓ℛ𝑔
𝐷 𝑓 ∘𝑔
ℛ 𝑓 ∘𝑔
Domain restriction
Functions & Graphs 17
Identity function𝑓 :𝑥⟼𝑥𝒟=ℝℛ=ℝ
Functions & Graphs 18
Inverse function
𝑓 −1 Is the inverse function of if:
( 𝑓 ∘ 𝑓 − 1 ) (𝑥 )= ( 𝑓 −1∘ 𝑓 ) (𝑥 )=𝑥
𝐷 𝑓 − 1=ℛ 𝑓
𝑅 𝑓 − 1=𝐷 𝑓
Only one-to-one functions are invertible !
Functions & Graphs 19
END
DisclaimerThis document is meant to be apprehended through professional teacher mediation (‘live in class’) together with a mathematics text book, preferably on IB level.