P.3 Functions and Their Graphs

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Calculus is something to
P.3
Functions and
Their Graphs
about!!!
Functions
Function - for every x there is exactly one y.
Domain - set of x-values
Range - set of y-values
Tell whether the equations represent y as a function
of x.
a.
x2 + y = 1
Solve for y.
y = 1 – x2
For every number we
plug in for x, do we get
more than one y out?
No, so this equation
is a function.
b.
-x + y2 = 1
Solve for y.
y2 = x + 1
Here we have 2 y’s for
each x that we plug in.
Therefore, this equation
is not a function.
y   1 x
Find the domain of each function.
a.
f: {(-3,0), (-1,4), (0,2), (2,2), (4,-1)}
Domain = { -3, -1, 0, 2, 4}
b.
c.
1
g ( x) 
x5
f ( x)  4  x
D:
2
(2  x)(2  x)  0
D:
[-2, 2]
, x  5
Set 4 – x2 greater
than or = to 0, then
factor, find C.N.’s
and test each interval.
Ex.
g(x) = -x2 + 4x + 1
Find: a.
g(2)
b.
g(t)
c.
g(x+2)
d.
g(x + h)
Ex.
 x  1, x  0
f ( x)  

 x  1, x  0 
2
Evaluate at x = -1, 0, 1
Ans. 2, -1, 0
Ex. f(x) = x2 – 4x + 7
Find.
f ( x  h)  f ( x )
h
[(x  h)  4( x  h)  7]  [ x  4 x  7]

h
2
2
2
x  2 xh  h  4 x  4h  7  x  4 x  7

h
2
h(2 x  h  4)
2 xh  h  4h

= 2x + h - 4

h
h
2
2
(2,4)
Find:
a. the domain
[-1,4)
b. the range
[-5,4]
c. f(-1) = -5
(4,0)
(-1,-5)
d. f(2) = 4
Day 1
Vertical Line Test for Functions
Do the graphs represent y as a function of x?
no
yes
yes
Tests for Even and Odd Functions
A function is y = f(x) is even if, for each x in the
domain of f,
f(-x) = f(x)
An even function is symmetric about the y-axis.
A function is y = f(x) is odd if, for each x in the
domain of f,
f(-x) = -f(x)
An odd function is symmetric about the origin.
Ex. g(x) = x3 - x
g(-x) = (-x)3 – (-x) = -x3 + x = -(x3 – x)
Therefore, g(x) is odd because f(-x) = -f(x)
Ex. h(x) = x2 + 1
h(-x) = (-x)2 + 1 = x2 + 1
h(x) is even because f(-x) = f(x)
Summary of Graphs of Common Functions
f(x) = c
y=x
y x
y x
y=x3
y = x2
Vertical and Horizontal Shifts
On calculator, graph y = x2
graph y = x2 + 2
y = x2 - 3
y = (x – 1)2
y = (x + 2)2
y = -x2
y = -(x + 3)2 -1
Vertical and Horizontal Shifts
1.
h(x) = f(x) + c
Vert. shift up
2.
h(x) = f(x) - c
Vert. shift down
3.
h(x) = f(x – c)
Horiz. shift right
4.
h(x) = f(x + c)
Horiz. shift left
5.
h(x) = -f(x)
Reflection in the x-axis
6.
h(x) = f(-x)
Reflection in the y-axis
Combinations of
Functions
The composition of the functions f and g is
( f  g )(x)  f ( g ( x))
“f composed by g of x equals f of g of x”
Ex. f(x) =
g(x) = x - 1
x
 f  g (2)
 f  g x  f ( g ( x)) 
Find
of 2
x 1
 2 1  1
Ex. f(x) = x + 2 and g(x) = 4 – x2
 f  g x  
Find:
f(g(x)) = (4 – x2) + 2
= -x2 + 6
g  f x  
g(f(x)) = 4 – (x + 2)2 = 4 – (x2 + 4x + 4)
= -x2 – 4x
1
Ex. Express h(x) =
as
a
composition
2
x  2
of two functions f and g.
f(x) =
1
2
x
g(x) =
x-2
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