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Lecture 3

Department

MathematicsCourse Code

MATA30H3Professor

Xiao JieLecture

3This

**preview**shows half of the first page. to view the full**3 pages of the document.**MATA30 Lecture 3 – Functions 2

Even & Odd Functions

- Even: ƒ(-n) = ƒ(n)

o Replace n with -n, simplify and get back ƒ(n)

o Vertically symmetric across y – axis

- Odd: ƒ(-n) = -ƒ(n)

o Replace n with -n, simplify & get back ƒ(n) with –

o Rotated 180˚ about the origin

- Neither: ƒ(-n) ≠ -ƒ(n)

≠ ƒ(n)

Trigonometric Even & Odd Functions

- Sin(x) is ODD → Sin(-x) = -sin(x)

- Cos(x) is EVEN → cos(-x) = cos(x)

- Tan(x) is ODD → tan(-x) = -tan(x)

y = tanx

Piecewise Defined Functions

- Define ƒ(n) in “pieces”

- Differently on different pieces of domain

→ absolute value |x|

|2| = 2

|-2| = 2 = -(-2)

ƒ(x) = |x| {𝑥 𝑖𝑓 𝑥 ≥ 0

−𝑥 𝑖𝑓 𝑥 < 0

Increasing/ Decreasing Functions

→ increasing

o For all x1 < x2, ƒ(n1) < ƒ(n2)

→ decreasing

o For all x1 < x2, ƒ(n1) > ƒ(n2)

y = sinx y = cosx

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