Functions&

Graphs

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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.