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Domain and range
Q&A
Can a function’s domain and range be the same?
Yes. For example, the domain and range of the cube root function are both the set of all real numbers.
Finding domains and ranges of the toolkit functions
We will now return to our set of toolkit functions to determine the domain and range of each.
For the
constant function
f
(
x
)
=
c
,
the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant
c
,
so the range is the set
{
c
}
that contains this single element. In interval notation, this is written as
[
c
,
c
]
,
the interval that both begins and ends with
c
.
For the
identity function
f
(
x
)
=
x
,
there is no restriction on
x
.
Both the domain and range are the set of all real numbers.
For the
absolute value function
f
(
x
)
=
|
x
|
,
there is no restriction on
x
.
However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.
For the
quadratic function
f
(
x
)
=
x
2
,
the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.
For the
cubic function
f
(
x
)
=
x
3
,
the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.
For the
reciprocal function
f
(
x
)
=
1
x
,
we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write
{
x
|
x
≠
0
}
, the set of all real numbers that are not zero.
For the
reciprocal squared function
f
(
x
)
=
1
x
2
, we cannot divide by
0
, so we must exclude
0 from the domain. There is also no
x that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.
For the
square root function
f
(
x
)
=
x
,
we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number
x
is defined to be positive, even though the square of the negative number
−
x
also gives us
x
.
For the
cube root function
f
(
x
)
=
x
3
,
the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).
How To
Given the formula for a function, determine the domain and range.
Exclude from the domain any input values that result in division by zero.
Exclude from the domain any input values that have nonreal (or undefined) number outputs.
Use the valid input values to determine the range of the output values.
Look at the function graph and table values to confirm the actual function behavior.
Questions & Answers
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studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
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Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
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the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
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Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
Lubega
Heyy Lubega hussein where are u from?
_Adnan
which site have a normal flora
Many sites of the body have it
Skin
Nasal cavity
Oral cavity
Gastro intestinal tract
Safaa
skin,Oral,Nasal,GIt
Sadik
How can Commensal can Bacteria change into pathogen?
Sadik
How can Commensal Bacteria change into pathogen?
Sadik
what are the advantages of normal Flora to the host
Micheal
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Micheal
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Wilfred
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Binomial nomenclature
adeolu
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Source:
OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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