Domain

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Domain
and
Range
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As we study functions
we learn terms like
input values
and
output values.
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Input values are the numbers
we put into the function.
They are the x-values.
Output values are the numbers
that come out of the function.
They are the y-values.
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Given the function,
we can choose any value we want
for x.
Suppose we choose 11.
We can put 11 into the function by
substituting for x.
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If we wrote down every number we could
put in for x and still have the function
make sense,
we would have the set of numbers we call the
domain of the function.
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The domain is the set that
contains all the input values
for a function.
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In our function
is there any number we could
not put in for x?
No!
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Because we could substitute
any real number
for x,
we say the domain
of the function
is the set of real numbers.
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To use the symbols of algebra,
we could write the domain as
Does that look like a foreign language?
Lets translate
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The curly braces
just tell us we have a set of numbers.
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The x reminds us
that our set contains x-values.
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The colon says,
such that
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The symbol that looks like an e
(or a c sticking its tongue out)
says, belongs to . . .
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And the cursive, or script,
R
is short for the set of real numbers.
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So we read it, The set
of x
such that
x belongs to
R, the set of real numbers.
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When we put 11 in for x,
y was 17.
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So 17 belongs to
the range of the function,
Is there any number that
we could not get for y by
putting some number in for x?
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No!
We say that the range of
the function is
the set of real numbers.
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Read this
The set of y, such that
y belongs to R,
the set of real numbers.
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It is not always true that
the domain and range
can be any real number.
Sometimes mathematicians
want to study a function over
a limited domain.
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They might think about
the function
where x is between 3 and 3.
It could be written,
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Sometimes the function itself
limits the domain or range.
In this function,
can x be any real number?
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What would happen if x
were 3?
We can never divide by
0.
Then we would have to divide by
0.
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So we would have to eliminate
3 from the domain.
The domain would be,
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Can you think of a number
which could not belong to the range?
y could never be 0.
Why?
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What would x have to be
for y to be 0?
There is no number we can divide 1 by to get 0,
so 0 cannot belong to the range.
The range of the function is,
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The most common rules of algebra
that limit the domain of functions are
Rule 1 You cant divide by 0.
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Weve already seen an example
of Rule 1 You cant divide by 0.
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Think about Rule 2,
You cant take the square root of a
negative number.
Given the function,
what is the domain?
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What is y when x is 16?
The square root of 16 is 4,
so y is 4 when x is 16
16 belongs to the domain,
and 4 belongs to the range.
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But what is y when x
is 16?
What number do you square to get 16?
Did you say 4?
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not 16.
There is no real number we can square to get a
negative number.
So no negative number can belong to the
domain of
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The smallest number for which we can find a
square root is 0,
so the domain of
is
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Find the domain of each function
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Answers