Functions

I. Definitions

-7 -6 -2 -4 Relation: A set of ordered pairs ( , ) 1 3 ( , ) 9 ( , ) 8 ( , ) 1 3 ( , ) 9 -7 ( , ) 8 5 ( , ) -6 -2 ( , ) -4

-7 -6 -2 -4 Relation: A set of ordered pairs ( , ) 1 3 ( , ) 9 ( , ) 8 ( , ) 1 3 ( , ) 9 -7 ( , ) 8 5 ( , ) -6 -2 ( , ) -4 Domain: The set of first terms of each pair.

-7 -6 -2 -4 Relation: A set of ordered pairs ( , ) 1 3 ( , ) 9 ( , ) 8 ( , ) 1 3 ( , ) 9 -7 ( , ) 8 5 ( , ) -6 -2 ( , ) -4 Range: The set of second terms of each pair.

Representations of Relations 1, 3 9,−7 8, 5 −9,−7 −4, 0 Ordered Pairs

Representations of Relations X Y 1 3 -7 9 8 5 -6 -2 -4 Table

Representations of Relations Graph

Representations of Relations 1 3 9 -7 8 5 -6 -2 -4 Mapping

Find the domain and range. 𝐷𝑜𝑚𝑎𝑖𝑛: 0 , ∞ 𝑅𝑎𝑛𝑔𝑒: 0 , ∞

Find the domain and range. 𝐷𝑜𝑚𝑎𝑖𝑛:(−∞, ∞)

Find the domain and range. 𝑦=−2 𝐷𝑜𝑚𝑎𝑖𝑛: 0 , ∞ 𝑅𝑎𝑛𝑔𝑒: −2 , ∞

Find the domain and range. 𝐷𝑜𝑚𝑎𝑖𝑛: −∞ , ∞ 𝑅𝑎𝑛𝑔𝑒: −∞ , ∞

Find the domain and range. 𝑦=2 𝑦=−3 𝐷𝑜𝑚𝑎𝑖𝑛: −∞ , ∞ 𝑅𝑎𝑛𝑔𝑒: −3, 2

Now it’s your turn

Find the domain and range. B. C. D.

Find the domain and range. 𝑥=−4 𝐷𝑜𝑚𝑎𝑖𝑛:−4 𝑅𝑎𝑛𝑔𝑒: −∞ , ∞

Find the domain and range. 𝑦=3 𝐷𝑜𝑚𝑎𝑖𝑛: −∞ , ∞ 𝑅𝑎𝑛𝑔𝑒:3

Find the domain and range. 𝐷𝑜𝑚𝑎𝑖𝑛: −∞,0 ∪ 0,∞ OR 𝑥≠0

Find the domain and range. 𝐷𝑜𝑚𝑎𝑖𝑛: −∞,0 ∪ 0,∞ OR 𝑥≠0 𝑅𝑎𝑛𝑔𝑒: −∞,0 ∪ 0,∞ OR 𝑦≠0

Find the domain and range. 𝐷𝑜𝑚𝑎𝑖𝑛: 0,∞ 𝑅𝑎𝑛𝑔𝑒: −∞,∞

II. Functions

-6 -5 -7 -2 Functions: A relation with no repeated domain members ( , ) 3 ( , ) 4 -6 ( , ) 1 9 ( , ) -5 -7 ( , ) -2 8 Function

-4 -5 -9 -8 Functions: A relation with no repeated domain members ( , ) 7 3 ( , ) -4 ( , ) 7 1 ( , ) -5 -9 ( , ) -8 1 Not a Function

The Vertical Line Test Function Not a Function

Now it’s your turn

Function or Naw?

Function or Naw?

Function or Naw?

Function or Naw?

Function or Naw?

Function or Naw?

` Function or Naw?

III. Evaluating Functions

𝑓 𝑥 =2𝑥+3 Evaluate and simplify 𝑓(−3)

𝑓 𝑥 =2𝑥+3 𝑓 −3 =2⋅ −3 +3 𝑓 −3 =−6+3 𝑓 −3 =−3

𝑔 𝑥 =− 𝑥 3 +3𝑥 Evaluate and simplify 𝑔(−2)

𝑔 𝑥 =− 𝑥 3 +3𝑥 𝑔 −2 =− −2 3 +3 −2 𝑔 −2 =− −8 −6

𝑔 −2 =8−6 𝑔 −2 =2 𝑔 −2 =− −8 −6

𝑔 𝑥 =− 𝑥 3 +3𝑥 𝑔 10 =− 10 3 +3⋅10 𝑔 10 =− 1000 +30 𝑔 10 =−970

ℎ 𝑥 = −3𝑥+6 Evaluate and simplify ℎ(5)

ℎ 𝑥 = −3𝑥+6 ℎ 5 = −3⋅5+6 ℎ 5 = −15+6

ℎ 5 = −9 ℎ 5 =9 ℎ 5 = −15+6

𝑦 𝑥 =𝑙𝑜𝑔 𝑥 Evaluate and simplify 𝑦(1000)

𝑦 𝑥 =𝑙𝑜𝑔 𝑥 𝑦 1000 =𝑙𝑜𝑔 1000 𝑦 1000 =3

𝑝 𝑥 = 4 𝑥 Evaluate and simplify 4 8

𝑝 𝑥 = 4 𝑥 𝑝 8 = 4 8 𝑝 8 =65536

Now it’s your turn

𝑓 𝑥 =2𝑥+3 𝑓 7 =2⋅7+3 𝑓 7 =14+3 𝑓 7 =17

𝑓 7 𝑔 8 ℎ −4 𝑦 0.0001 𝑝 −3

𝑓 𝑥 =2𝑥+3 𝑓 7 =2⋅7+3 𝑓 7 =14+3

𝑓 7 =17 𝑓 7 =14+3

𝑔 𝑥 =− 𝑥 3 +3𝑥 𝑔 8 =− 8 3 +3 8 𝑔 8 =−512+24

𝑔 8 =−488 𝑔 8 =−512+24

ℎ 𝑥 = −3𝑥+6 ℎ −4 = −3⋅−4+6 ℎ −4 = 12+6

ℎ −4 = 18 ℎ −4 = 12+6 ℎ −4 =18

ℎ −4 = 18 ℎ −4 = 18 ℎ −4 = 12+6

𝑦 𝑥 =𝑙𝑜𝑔 𝑥 𝑦 0.0001 =𝑙𝑜𝑔 0.0001 𝑦 0.0001 =−4

𝑝 𝑥 = 4 𝑥 𝑝 −3 = 4 −3 𝑝 −3 =0.015625

IV. Maximums & Minimums

Relative Maximum: The largest function (y) value across a finite domain Relative Minimum: The smallest function (y) value across a finite domain

Relative Maximum: Relative Minimum:

Relative Maximum:

Relative Minimum:

V. Odd & Even Functions

The function 𝑓 𝑥 is an 𝐨𝐝𝐝 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 if: 𝑓 −𝑥 =−𝑓 𝑥

The function 𝑓 𝑥 is an 𝐞𝐯𝐞𝐧 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 if: 𝑓 −𝑥 =𝑓 𝑥