Calculus is something to

about!!!

P.3Functions andTheir 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 functionof x.

a. x2 + y = 1 Solve for y.

y = 1 – x2 For every number weplug in for x, do we getmore than one y out?No, so this equation

is a function.

b. -x + y2 = 1 Solve for y.

y2 = x + 1

xy +±= 1

Here we have 2 y’s for each x that we plug in.Therefore, this equationis not a function.

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.

5

1)(

+=x

xg D: 5, −≠ℜ x

c. 24)( xxf −=Set 4 – x2 greaterthan or = to 0, thenfactor, find C.N.’s and test each interval.0)2)(2( ≥+− xx

D: [-2, 2]

Ex. g(x) = -x2 + 4x + 1

Find: a. g(2)

b. g(t)

c. g(x+2)

d. g(x + h)

Ex.

⎭⎬⎫

⎩⎨⎧

≥−<+

=0,1

0,1)(

2

xx

xxxf

Evaluate at x = -1, 0, 1

Ans. 2, -1, 0

Ex. f(x) = x2 – 4x + 7

Find.

h

xfhxf )()( −+

h

xxhxhx ]74[]7)(4)[( 22 +−−++−+=

h

xxhxhxhx 747442 222 −+−+−−++=

h

hhxh 42 2 −+=

h

hxh )42( −+= = 2x + h - 4

(-1,-5)

(2,4)

(4,0)Find:a. the domain

b. the range

c. f(-1) =

d. f(2) =

[-1,4)

[-5,4]-5

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)

A function is y = f(x) is odd if, for each x in the domain of f,

f(-x) = -f(x)

An even function is symmetric about the y-axis.

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 = xxy =

xy = y = x2y = x 3

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

))(())(( xgfxgf =o“f composed by g of x equals f of g of x”

Ex. f(x) = x g(x) = x - 1

Find ( ) )2(gf o

( )( ) == ))(( xgfxgf o 1−x

of 2 112 =−=Ex. f(x) = x + 2 and g(x) = 4 – x2 Find:

( )( )

( )( )=

=

xfg

xgf

o

o f(g(x)) = (4 – x2) + 2

= -x2 + 6

g(f(x)) = 4 – (x + 2)2 = 4 – (x2 + 4x + 4)= -x2 – 4x

Ex. Express h(x) = as a composition

of two functions f and g. ( )22

1

−x

f(x) =

g(x) = x - 2

2

1

x