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
y
11 10
3
4
5
8
5
1
Is it a function?
1) x   y  5
4) y  4  x
2) x  y  3
5) x  4
2
3) y  x  5
Find the domain of each
1) f : (3,0),(1, 4),(0, 2),(2, 2),(4, 1)
1
2) h( x) 
x4
3) f ( x)  2 x 2  4
Find the domain of each
4) m( x)  4  3 x
5) m( x)  16  x 2
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
x2  1
g ( x) 
2x
3) Using your calculator approximate the
extrema for y = |x – 3| + |x + 4| - |x+2|
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
3) f(x) = x3 – 1
2) h(x) = x2 + 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?