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Precalculus
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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
Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.
taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.
Faith
While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma
Kamara
which drugs can we use for ulcers
Omeprazole
Cimetidine / Tagament
For the complicated once ulcer - kit
Patrick
what is the function of lymphatic system
to drain extracellular fluid all over the body.
asegid
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Adama
Anatomy is the identification and description of the structures of living things
Kamara
what's the difference between anatomy and physiology
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
AI-Robot
what is enzymes all about?
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
how does the stomach protect itself from the damaging effects of HCl
little girl okay how does the stomach protect itself from the damaging effect of HCL
Wulku
it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
Kamara
function of digestive system
function of digestive
Ali
what is the normal body temperature
please why 37 degree selcius normal temperature
Mark
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body
the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
Stephanie
anaemia is the decrease in RBC count hemoglobin count and PVC count
Eniola
what is the pH of the vagina
how does Lysin attack pathogens
Diya
I information on anatomy position and digestive system and there enzyme
anatomy of the female external genitalia
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Source:
OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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