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
$\text{\hspace{0.17em}}f(x)=c,\text{\hspace{0.17em}}$ the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant
$\text{\hspace{0.17em}}c,\text{\hspace{0.17em}}$ so the range is the set
$\text{\hspace{0.17em}}\left\{c\right\}\text{\hspace{0.17em}}$ that contains this single element. In interval notation, this is written as
$\text{\hspace{0.17em}}[c,c],\text{\hspace{0.17em}}$ the interval that both begins and ends with
$\text{\hspace{0.17em}}c.$
For the
identity function
$\text{\hspace{0.17em}}f(x)=x,\text{\hspace{0.17em}}$ there is no restriction on
$\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ Both the domain and range are the set of all real numbers.
For the
absolute value function
$\text{\hspace{0.17em}}f(x)=\left|x\right|,\text{\hspace{0.17em}}$ there is no restriction on
$\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ 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
$\text{\hspace{0.17em}}f(x)={x}^{2},\text{\hspace{0.17em}}$ 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
$\text{\hspace{0.17em}}f(x)={x}^{3},\text{\hspace{0.17em}}$ 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
$\text{\hspace{0.17em}}f(x)=\frac{1}{x},\text{\hspace{0.17em}}$ 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
$\left\{x\right|\text{}x\ne 0\},$ the set of all real numbers that are not zero.
For the
reciprocal squared function
$\text{\hspace{0.17em}}f(x)=\frac{1}{{x}^{2}},$ 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
$\text{\hspace{0.17em}}f(x)=\sqrt[]{x},\text{\hspace{0.17em}}$ 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
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is defined to be positive, even though the square of the negative number
$\text{\hspace{0.17em}}-\sqrt{x}\text{\hspace{0.17em}}$ also gives us
$\text{\hspace{0.17em}}x.$
For the
cube root function
$\text{\hspace{0.17em}}f(x)=\sqrt[3]{x},\text{\hspace{0.17em}}$ 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
nothing up todat yet

Miranda

aap konsi country se ho

jai

which language is that

Miranda

what is the formula for calculating algebraic

I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it

Miranda

state and prove Cayley hamilton therom

the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.

Miranda

algebra is a branch of the mathematics to calculate expressions follow.

Miranda

Miranda Drice would you mind teaching me mathematics? I think you are really good at math.
I'm not good at it. In fact I hate it. 😅😅😅

Jeffrey

lolll who told you I'm good at it

Miranda

something seems to wispher me to my ear that u are good at it. lol

Jeffrey

lolllll if you say so

Miranda

but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.

Jeffrey

which grade are you in though

Miranda

oh woww I understand

Miranda

haha. already finished college

Jeffrey

how about you? what grade are you now?

Jeffrey

I'm going to 11grade

Miranda

how come you finished in college and you don't like math though

Miranda

gotta practice, holmie

Steve

if you never use it you won't be able to appreciate it

Steve

I don't know why. But Im trying to like it.

Jeffrey

yes steve. you're right

Jeffrey

what is the solution of the given equation?

please where is the equation

Miranda

answer and questions in exercise 11.2 sums

how do u calculate inequality of irrational number?

Alaba

and I will walk you through it

Chris

cos (-z)= cos z .

Swadesh

what is the identity of 1-cos²5x equal to?

so is their any Genius in mathematics here let chat guys and get to know each other's

SORIE

okay no problem since we gather here and get to know each other

SORIE

hi im stupid at math and just wanna join here

Yaona

lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together

SORIE

what is the function of sine with respect of cosine , graphically

sinx sin2x is linearly dependent

The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1

Shemmy

Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1

Jeza

each term in a sequence below is five times the previous term what is the eighth term in the sequence

I don't understand how radicals works pls

How look for the general solution of a trig function

Source:
OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

Google Play and the Google Play logo are trademarks of Google Inc.