# 3.2 Domain and range  (Page 6/11)

 Page 6 / 11

## Finding the domain and range using toolkit functions

Find the domain and range of $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{3}-x.$

There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.

The domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)\text{\hspace{0.17em}}$ and the range is also $\text{\hspace{0.17em}}\left(-\infty ,\infty \right).$

## Finding the domain and range

Find the domain and range of $\text{\hspace{0.17em}}f\left(x\right)=\frac{2}{x+1}.$

We cannot evaluate the function at $\text{\hspace{0.17em}}-1\text{\hspace{0.17em}}$ because division by zero is undefined. The domain is $\text{\hspace{0.17em}}\left(-\infty ,-1\right)\cup \left(-1,\infty \right).\text{\hspace{0.17em}}$ Because the function is never zero, we exclude 0 from the range. The range is $\text{\hspace{0.17em}}\left(-\infty ,0\right)\cup \left(0,\infty \right).$

## Finding the domain and range

Find the domain and range of $\text{\hspace{0.17em}}f\left(x\right)=2\sqrt{x+4}.$

We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.

The domain of $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left[-4,\infty \right).$

We then find the range. We know that $\text{\hspace{0.17em}}f\left(-4\right)=0,\text{\hspace{0.17em}}$ and the function value increases as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ increases without any upper limit. We conclude that the range of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left[0,\infty \right).$

Find the domain and range of $\text{\hspace{0.17em}}f\left(x\right)=-\sqrt{2-x}.$

domain: $\text{\hspace{0.17em}}\left(-\infty ,2\right];\text{\hspace{0.17em}}$ range: $\text{\hspace{0.17em}}\left(-\infty ,0\right]$

## Graphing piecewise-defined functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function $\text{\hspace{0.17em}}f\left(x\right)=|x|.\text{\hspace{0.17em}}$ With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude , or modulus , of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.

If we input 0, or a positive value, the output is the same as the input.

$f\left(x\right)=x\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x\ge 0$

If we input a negative value, the output is the opposite of the input.

$f\left(x\right)=-x\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x<0$

Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function    is a function in which more than one formula is used to define the output over different pieces of the domain.

We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to \$10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income $\text{\hspace{0.17em}}S\text{\hspace{0.17em}}$ would be $\text{\hspace{0.17em}}0.1S\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}S\le \text{}10\text{,}000\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\text{}1000+0.2\left(S-\text{}10\text{,}000\right)\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}S>\text{}10\text{,}000.$

## Piecewise function

A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:

In piecewise notation, the absolute value function is

Given a piecewise function, write the formula and identify the domain for each interval.

1. Identify the intervals for which different rules apply.
2. Determine formulas that describe how to calculate an output from an input in each interval.
3. Use braces and if-statements to write the function.

what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
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jai
which language is that
Miranda
I am living in india
jai
good
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
hello
Propessor
hi
Miranda
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
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
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
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Miranda
Jeffrey
Jeffrey
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
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Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
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Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
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SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
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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
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
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