Functions(but not trig functions!)

Objectives: Be able to… -Identify, evaluate and find the domain of functions.-Determine where a graph is increasing, decreasing or constant-Find the extrema of a function-Determine if a function is even, odd or neither

TS: Make decisions after reflection and review

Formal Definition

A function, f, from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is the domain (or set of inputs) and the set B contains the range (or the set of outputs).

Is it a function? What is the domain & range?

Is it a function? What is the domain & range?

x 2 2 3 4 5

y 11 10 8 5 1

Is it a function?

4) 4y x

3) 5y x

1) 5x y

22) 3x y 5) 4x

Find the domain of each 1) : ( 3,0), ( 1,4), (0,2), (2,2), (4, 1)f

23) ( ) 2 4f x x

12) ( )

4h x

x

Find the domain of each

4) ( ) 4 3m x x

25) ( ) 16m x x

Increasing, Decreasing or Constant

Increasing Interval:

For any x1 and x2 in the interval x1 < x2 implies f(x1) < f(x2)

Decreasing Interval:

For any x1 and x2 in the interval x1 < x2 implies f(x1) > f(x2)

Constant Interval:

For any x1 and x2 in the interval x1 < x2 implies f(x1) = f(x2)

Find the open intervals of x over which the functions are increasing,

decreasing or constant. y=|x2 – 4|

Extrema (Absolute & Relative Maximums & Minimums)

1) Using your calculator approximate the extrema for f(x) = -x3 + x

2) Using your calculator approximate the extrema for

3) Using your calculator approximate the extrema for y = |x – 3| + |x + 4| - |x+2|

2 1( )

2

xg x

x

Even & Odd Functions

Even functions:

• Functions which have y-axis symmetry

• f(-x) = f(x)

Odd Functions:

• Functions which have origin symmetry

• f(-x) = – f(x)

Test algebraically to see if each function is even, odd or neither.

Then verify graphically.1) g(x) = x3 – x 2) h(x) = x2 + 1

3) f(x) = x3 – 1

Closure: See if you can think of answers to these two questions.

1) Find two non-polynomial even functions. They can’t both use the same parent.

2) Draw a picture of an object that has both origin and y-axis symmetry. Can you make one that is a function?