P.3
Functions and
Their Graphs

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.
x 2 + y = 1 y = 1 – x 2
No, so this equation is a function.
b.
-x + y 2 = 1 y 2 = x + 1 y
 
1
 x
Solve for y.
For every number we plug in for x, do we get more than one y out?
Solve for y.
Here we have 2 y’s for each x that we plug in.
Therefore, this equation is not a function.

Find the domain of each function.
a.
f: {(-3,0), (-1,4), (0,2), (2,2), (4,-1)} b. c.
Domain = { -3, -1, 0, 2, 4} g ( x )
 x
1

5 f ( x )

4
 x
2
( 2
 x )( 2
 x )

0
D:

, x
 
5
Set 4 – x 2 greater than or = to 0, then factor, find C.N.’s and test each interval.
D: [-2, 2]

Ex.
g(x) = -x 2 + 4x + 1
Find: a.
g(2) b.
g(t) c.
g(x+2) d.
g(x + h)
Ex.
f ( x )



 x x
2


1 ,
1 , x x


0
0



Evaluate at x = -1, 0, 1
Ans. 2, -1, 0

Ex. f(x) = x 2 –
4x + 7
Find.
f ( x
 h )
 f ( x )


[( x
 h )
2 h

4 ( x
 h )

7 ]

[ x
2 x
2 
2 xh
 h
2

4 x

7 ]

4 x
 h
4 h

7
 x
2 
4 x

7

2 xh
 h h
2 
4 h
 h h ( 2 x
 h

4 ) h
= 2x + h - 4

Find: a. the domain
[-1,4) b. the range
[-5,4] c. f(-1) = -5 d. f(2) = 4
(-1,-5)
Day 1
(2,4)
(4,0)

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) = x 3 - x g(-x) = (-x) 3 – (-x) = -x 3 + x = -(x 3 – x)
Therefore, g(x) is odd because f(-x) = -f(x)
Ex. h(x) = x 2 + 1 h(-x) = (-x) 2 + 1 = x 2 + 1 h(x) is even because f(-x) = f(x)

y
Summary of Graphs of Common Functions f(x) = c y = x y
 x
 x y = x 2 y = x 3

Vertical and Horizontal Shifts
On calculator, graph y = x 2 graph y = x 2 + 2 y = x 2 - 3 y = (x – 1) 2 y = (x + 2) 2 y = -x 2 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)
6.
h(x) = f(-x)
Reflection in the x-axis
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) =
Find
 f  g
  x
 f  g

( 2 ) g(x) = x - 1
 f ( g ( x ))
 x

1 of 2

2

1

1
Ex. f(x) = x + 2 and g(x) = 4 – x 2 Find:
 f  g
 
 f(g(x)) = (4 – x 2 ) + 2
= -x 2 + 6
 g  f
 
 g(f(x)) = 4 – (x + 2) 2 = 4 – (x 2 + 4x + 4)
= -x 2 – 4x

 x

1
2

2 of two functions f and g. f(x) =
1 x
2 g(x) = x - 2