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When a function is transformed, its domain and/or range will change. If
only the inputs are transformed, then only the domain will change. If only
the outputs are transformed, then only the range will change. If both the
inputs and outputs are transformed, then both the domain and range will
change.
Remember that the domain represents the set of inputs for a function,
and the range represents the set of outputs.
Example 1 : Let ๐ฆ = ๐
be a function with domain ๐ท =
[
]
and
range ๐
=
[
]
. Find the domain ๐ท and range ๐
for each of the
following functions. Keep in mind order of operation and the order of
your intervals.
a. ๐ฆ = โ 3 ๐
b. ๐ฆ = ๐ (
1
2
Changes IN side the parentheses change the IN puts and we do the
IN verse; remember that the Domain is the set of inputs
Changes OUT side the parentheses change the OUT puts and we do
exactly what we see; remember that the Range is the set of outputs
inputs, โ 3
outputs
inputs
1
2
, outputs
Range:
[
)]
Domain: [โ
6
1
2
5
1
2
]
Range:
[
]
Domain:
[
]
[
]
[
]
[
]
[
]
Since the inputs of this function are not
being changed with the transformation
๐ฆ = โ 3 ๐
( ๐ฅ
) , that means the domain is
also not being changed. So the domain
will still be ๐ท = [โ 6 , 5 ].
Since the outputs of this function are
not being changed with the
transformation ๐ฆ = ๐ (
1
2
๐ฅ), that means
the range is also not being changed. So
the range will remain ๐ท = [ 0 , 14 ].
Be sure to keep in mind that intervals (such as a domain or range),
just like number lines, always go in order from smallest to largest as
you go from left to right. On Example 1a, the range is listed as
[
]
because โ๐๐ is smaller than ๐.
Be sure to re-arrange intervals as needed so they are in the correct
order.
Example 2 : Let ๐ฆ = ๐(๐ฅ) be a function with domain ๐ท =
[
]
and
range ๐
= [ 0 , 14 ]. Find the domain ๐ท and range ๐
for each of the
following functions. Keep in mind order of operation and the order of
your intervals.
a. ๐ฆ = ๐
โ 2 b. ๐ฆ = ๐
b.
Example 3 : Let ๐ฆ = ๐
be a function with domain ๐ท =
[
]
and
range ๐
=
[
]
. Find the domain ๐ท and range ๐
for each of the
following functions. Keep in mind order of operation and the order of
your intervals.
a. ๐ฆ =
1
2
b. ๐ฆ = โ๐
b.
Example 6 : Let ๐ฆ = ๐
be a function with domain ๐ท =
[
and
range ๐
=
]
. Find the domain ๐ท and range ๐
for each of the
following functions. Keep in mind order of operation and the order of
your intervals.
a. ๐ฆ =
2
3
โ 1 b. ๐ฆ = โ 3 ๐ (โ
1
3
Example 7 : Let ๐ฆ = ๐(๐ฅ) be a function with domain ๐ท = [โ 9 , 0 ] and
range ๐
=
. Find the domain ๐ท and range ๐
for each of the
following functions. Keep in mind order of operation and the order of
your intervals.
a. ๐ฆ = ๐
b. a
inputs + 4 , outputs + 1
inputs
3
outputs
Domain:
[
]
Domain: [โ
9
3
0
3
]
Range:
Range: (โ 1
Domain:
[
]
Domain:
[
]
Range:
Range:
[
]
[
]
Example 8 : Let ๐ฆ = ๐
be a function with domain ๐ท =
and
range ๐
=
[
]
. Find the domain ๐ท and range ๐
for each of the
following functions. Keep in mind order of operation and the order of
your intervals.
a. ๐ฆ =
1
2
Once again, keep in mind that the domain of a function is the set of inputs,
while the range of a function is the set of outputs. So any changes to the
inputs of a function are made to the domain, and any changes to the
outputs of a function are made to the range.
Answers to Examples:
1 a. ๐ท:
[
]
[
]
; 1 b. ๐ท:
[
]
[
]
2a. ๐ท:
[
]
[
]
; 2b. :
[
]
[
]
3a. :
[
]
[
]
; 3b. ๐ท: [โ 2 ,
5
3
] , ๐
:
[
]
4a. :
[
]
, ๐
: [โ 1 ,
25
3
] ; 4b. ๐ท: [โ
10
3
, 4 ] , ๐
:
[
]
5 a. ๐ท: (โโ, 0 ], ๐
: (โโ, 3 ] ; 5 b. ๐ท: [ 0 , โ), ๐
: [โ 2 , โ) ;
6a. ๐ท:
[
]
; 6b. ๐ท:
]
[
7 a. ๐ท:
[
]
; 7 b. ๐ท:
[
]
8 a. ๐ท: (โโ, โ), ๐
: [
1
2
, 5 ] ; 8 b. ๐ท: (โโ, โ), ๐
: [โ 22 , 23 ] ;
