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Unit 5 Seminar:

Functions and Graphs

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Line y = 2x + 1

Function f(x) = 2x + 1

x -------------> ---------> f(x) f

Evaluating functions

Ex: g(x) = 3x + 2

Then g(0) = g(1) = g(b) = g(x+4) =

Different types of Functions

f(x) = x2 + 1

f(x) = | x |

Different types of Functions

Piecewise functions

Ex: f(x) = x2 when x <0 = 5x -1 when 0 ≤ x < 3

= 2x when x ≥ 3

What is f(-1), f(2), f(3)?

Relation

a relation is a set of ordered pairs

Ex: { (1,0), (3,2), (1,5), (6,8), (7,2) }

Domain is the set of first elements {1, 3, 6, 7}Range is the set of second elements {0, 2, 5, 8}

Functions are special relations

Ex: { (1,2), (1, 3), (3, 4), (7,8) } not a function

Ex: { (1,3), (2,3), (5,7), (6,7) } function?

y = x2 is a function

every x goes to one y

x = y2 does not define y as a function of x

one x goes to two y's

f(x) = x4 - 5x2 + 4

Odd and even functions

A function is even is f(-x) = f(x) ex: f(x) = x2

A function is odd if f(-x) = -f(x) ex: f(x) = x3

A function is neither of none of the above apply

g(x) = x5 - x odd or even or neither?

f(x) = | x |

g(x) = f(x+2) = | x + 2 |

g(x) = f(x - 2) = | x - 2 |

g(x) = f(x) + 1 = | x | + 1

g(x) = f(x) - 1 = | x | - 1

g(x) = 3f(x) = 3| x |

Composition of functions

(f o g)(x) = f( g(x) )

x ----> ----> g(x) -----> ---> f(g(x)) g f

Ex: let f(x) = 3x + 1 g(x) = x2

Find (f o g)(x)

Ex: let f(x) = 3 - x g(x) = x3

Find (g o f)(2)

Inverse functions

Sometimes you get

x ----> ---------> ---> x

f "undid" g

g f

If both

(f o g)(x) = x (g o f)(x) = x

Then g is the inverse of f and we write g = f-1

Find the inverse of f(x) = 3x - 2