Helpful

Notes:

Finding

l)omain

& Range

Ilomain:

Possible

values that

can be used

in

a

function.

There are

situations

where certain x-values

can

not be used

as

inputs

(i.e.,

for f(x)d(x-l), x:.1

is not

in the domain since

it makes the

bottom

zero

while the

top

is I

(no

outputs

or

y-values

since it

makes

for an undefined situation.

Another

example

would be

f(x):Ji.

The inputs

must be

greater

than or equal

to zero

(x20)

since the square

root of a

negative

number

can not be

taken

(no

outputs)

for real

valued functions.'

Yisual way

for frnding

the domain:

Look at the

graph

and

imagine all

possible

vertical lines

(these

are

x-values)

that

intersect

the curve.

Range:

For a

given

domain,

these

will be all

possible

outputs

(y-values

or functional

values)

Visual way

for

finding the

range: Since

an output

(y-value)

is the

distance

from

the x-axis to the

curye

(if

curve

is below the

x-axis,

it will

be negative), estimate

this distance.

Since

y-values

are

the

same along

any horizontal,

you

can

get

many on

the

y-axis

since it

is

usually calibrated.

In

some cases,

you'll

have

to

estimate

them

if the

graph

is

given.

Piecewise

defined

functions

(like

the sketches

in lesson #1):

To sketch

ttrem,

look at the far

right

where the x-values

are

given.

There could be a number of lines

here

indicating

different

sections

or

values for x and

to the left of them

the outputs

(y-values)

for

these

inputs

(x-values).

The domain

would comprise

all the

x-values for all the lines of definition.

Make sure

you

use the

proper

y-values

for that

line in the definition.

The most common mistake

would be using

allof

the curye

instead of erasing

the

part

you

can't use.

'

To check

yourself,

look at

the competed

graph

and see if

all

vertical

lines that intersect

the curve

only do so once

and only

once.

Below

is an illustration

of this check.

Also, the competed curve

is considered

ONE function,

reg#dless

of the

number of

lines of definition on

the right.

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=

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