# 3.2 Domain and range  (Page 8/11)

 Page 8 / 11

## Verbal

Why does the domain differ for different functions?

The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.

How do we determine the domain of a function defined by an equation?

Explain why the domain of $\text{\hspace{0.17em}}f\left(x\right)=\sqrt[3]{x}\text{\hspace{0.17em}}$ is different from the domain of $\text{\hspace{0.17em}}f\left(x\right)=\sqrt[]{x}.$

There is no restriction on $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}f\left(x\right)=\sqrt[3]{x}\text{\hspace{0.17em}}$ because you can take the cube root of any real number. So the domain is all real numbers, $\text{\hspace{0.17em}}\left(-\infty ,\infty \right).\text{\hspace{0.17em}}$ When dealing with the set of real numbers, you cannot take the square root of negative numbers. So $\text{\hspace{0.17em}}x$ -values are restricted for $\text{\hspace{0.17em}}f\left(x\right)=\sqrt[]{x}\text{\hspace{0.17em}}$ to nonnegative numbers and the domain is $\text{\hspace{0.17em}}\left[0,\infty \right).$

When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?

How do you graph a piecewise function?

Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the $\text{\hspace{0.17em}}x$ -axis and $\text{\hspace{0.17em}}y$ -axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate $\text{\hspace{0.17em}}-\infty \text{\hspace{0.17em}}$ or Combine the graphs to find the graph of the piecewise function.

## Algebraic

For the following exercises, find the domain of each function using interval notation.

$f\left(x\right)=-2x\left(x-1\right)\left(x-2\right)$

$f\left(x\right)=5-2{x}^{2}$

$\left(-\infty ,\infty \right)$

$f\left(x\right)=3\sqrt{x-2}$

$f\left(x\right)=3-\sqrt{6-2x}$

$\left(-\infty ,3\right]$

$f\left(x\right)=\sqrt{4-3x}$

$\begin{array}{l}\\ f\left(x\right)=\sqrt[]{{x}^{2}+4}\end{array}$

$\left(-\infty ,\infty \right)$

$f\left(x\right)=\sqrt[3]{1-2x}$

$f\left(x\right)=\sqrt[3]{x-1}$

$\left(-\infty ,\infty \right)$

$f\left(x\right)=\frac{9}{x-6}$

$f\left(x\right)=\frac{3x+1}{4x+2}$

$\left(-\infty ,-\frac{1}{2}\right)\cup \left(-\frac{1}{2},\infty \right)$

$f\left(x\right)=\frac{\sqrt{x+4}}{x-4}$

$f\left(x\right)=\frac{x-3}{{x}^{2}+9x-22}$

$\left(-\infty ,-11\right)\cup \left(-11,2\right)\cup \left(2,\infty \right)$

$f\left(x\right)=\frac{1}{{x}^{2}-x-6}$

$f\left(x\right)=\frac{2{x}^{3}-250}{{x}^{2}-2x-15}$

$\left(-\infty ,-3\right)\cup \left(-3,5\right)\cup \left(5,\infty \right)$

$\frac{5}{\sqrt{x-3}}$

$\frac{2x+1}{\sqrt{5-x}}$

$\left(-\infty ,5\right)$

$f\left(x\right)=\frac{\sqrt{x-4}}{\sqrt{x-6}}$

$f\left(x\right)=\frac{\sqrt{x-6}}{\sqrt{x-4}}$

$\left[6,\infty \right)$

$f\left(x\right)=\frac{x}{x}$

$f\left(x\right)=\frac{{x}^{2}-9x}{{x}^{2}-81}$

$\left(-\infty ,-9\right)\cup \left(-9,9\right)\cup \left(9,\infty \right)$

Find the domain of the function $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{2{x}^{3}-50x}\text{\hspace{0.17em}}$ by:

1. using algebra.
2. graphing the function in the radicand and determining intervals on the x -axis for which the radicand is nonnegative.

## Graphical

For the following exercises, write the domain and range of each function using interval notation.

domain: $\text{\hspace{0.17em}}\left(2,8\right],\text{\hspace{0.17em}}$ range $\text{\hspace{0.17em}}\left[6,8\right)\text{\hspace{0.17em}}$

domain: range:

domain: range: $\text{\hspace{0.17em}}\left[0,2\right]$

domain: $\text{\hspace{0.17em}}\left(-\infty ,1\right],\text{\hspace{0.17em}}$ range: $\text{\hspace{0.17em}}\left[0,\infty \right)\text{\hspace{0.17em}}$

domain: $\text{\hspace{0.17em}}\left[-6,-\frac{1}{6}\right]\cup \left[\frac{1}{6},6\right];\text{\hspace{0.17em}}$ range: $\text{\hspace{0.17em}}\left[-6,-\frac{1}{6}\right]\cup \left[\frac{1}{6},6\right]\text{\hspace{0.17em}}$

domain: range: $\text{\hspace{0.17em}}\left[0,\infty \right)\text{\hspace{0.17em}}$

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

$f\left(x\right)=\left\{\begin{array}{lll}x+1\hfill & \text{if}\hfill & x<-2\hfill \\ -2x-3\hfill & \text{if}\hfill & x\ge -2\hfill \end{array}$

$f\left(x\right)=\left\{\begin{array}{lll}2x-1\hfill & \text{if}\hfill & x<1\hfill \\ 1+x\hfill & \text{if}\hfill & x\ge 1\hfill \end{array}$

domain: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$

$f\left(x\right)=\left\{\begin{array}{c}x+1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x<0\\ x-1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x>0\end{array}$

$f\left(x\right)=\left\{\begin{array}{ccc}3& \text{if}& x<0\\ \sqrt{x}& \text{if}& x\ge 0\end{array}$

domain: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$

$f\left(x\right)=\left\{\begin{array}{r}\hfill \begin{array}{r}\hfill {x}^{2}\\ \hfill x+2\end{array}\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{l}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x<0\hfill \\ \text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\ge 0\hfill \end{array}$

domain: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$

$f\left(x\right)=\left\{\begin{array}{ccc}x+1& \text{if}& x<1\\ {x}^{3}& \text{if}& x\ge 1\end{array}$

$f\left(x\right)=\left\{\begin{array}{c}|x|\\ 1\end{array}\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x<2\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\ge 2\hfill \end{array}$

domain: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$

## Numeric

For the following exercises, given each function $f,$ evaluate $f\left(-3\right),\text{\hspace{0.17em}}f\left(-2\right),\text{\hspace{0.17em}}f\left(-1\right),$ and $f\left(0\right).$

$f\left(x\right)=\left\{\begin{array}{lll}x+1\hfill & \text{if}\hfill & x<-2\hfill \\ -2x-3\hfill & \text{if}\hfill & x\ge -2\hfill \end{array}$

$\begin{array}{cccc}f\left(-3\right)=1;& f\left(-2\right)=0;& f\left(-1\right)=0;& f\left(0\right)=0\end{array}$

For the following exercises, given each function $\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}$ evaluate $f\left(-1\right),\text{\hspace{0.17em}}f\left(0\right),\text{\hspace{0.17em}}f\left(2\right),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(4\right).$

$f\left(x\right)=\left\{\begin{array}{lll}7x+3\hfill & \text{if}\hfill & x<0\hfill \\ 7x+6\hfill & \text{if}\hfill & x\ge 0\hfill \end{array}$

$\begin{array}{cccc}f\left(-1\right)=-4;& f\left(0\right)=6;& f\left(2\right)=20;& f\left(4\right)=34\end{array}$

$f\left(x\right)=\left\{\begin{array}{ccc}{x}^{2}-2& \text{if}& x<2\\ 4+|x-5|& \text{if}& x\ge 2\end{array}$

$f\left(x\right)=\left\{\begin{array}{ccc}5x& \text{if}& x<0\\ 3& \text{if}& 0\le x\le 3\\ {x}^{2}& \text{if}& x>3\end{array}$

$\begin{array}{cccc}f\left(-1\right)=-5;& f\left(0\right)=3;& f\left(2\right)=3;& f\left(4\right)=16\end{array}$

For the following exercises, write the domain for the piecewise function in interval notation.

domain: $\text{\hspace{0.17em}}\left(-\infty ,1\right)\cup \left(1,\infty \right)$

$f\left(x\right)=\left\{\begin{array}{c}2x-3\\ -3{x}^{2}\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x<0\\ \text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\ge 2\end{array}$

## Technology

Graph $\text{\hspace{0.17em}}y=\frac{1}{{x}^{2}}\text{\hspace{0.17em}}$ on the viewing window $\text{\hspace{0.17em}}\left[-0.5,-0.1\right]\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left[0.1,0.5\right].\text{\hspace{0.17em}}$ Determine the corresponding range for the viewing window. Show the graphs.

window: $\text{\hspace{0.17em}}\left[-0.5,-0.1\right];\text{\hspace{0.17em}}$ range:

window: range:

Graph $\text{\hspace{0.17em}}y=\frac{1}{x}\text{\hspace{0.17em}}$ on the viewing window $\text{\hspace{0.17em}}\left[-0.5,-0.1\right]\text{\hspace{0.17em}}$ and Determine the corresponding range for the viewing window. Show the graphs.

## Extension

Suppose the range of a function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is What is the range of $\text{\hspace{0.17em}}|f\left(x\right)|?$

Create a function in which the range is all nonnegative real numbers.

Create a function in which the domain is $\text{\hspace{0.17em}}x>2.$

Many answers. One function is $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{\sqrt{x-2}}.$

## Real-world applications

The height $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ of a projectile is a function of the time $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ it is in the air. The height in feet for $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ seconds is given by the function $h\left(t\right)=-16{t}^{2}+96t.$ What is the domain of the function? What does the domain mean in the context of the problem?

The domain is it takes 6 seconds for the projectile to leave the ground and return to the ground

The cost in dollars of making $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ items is given by the function $\text{\hspace{0.17em}}C\left(x\right)=10x+500.$

1. The fixed cost is determined when zero items are produced. Find the fixed cost for this item.
2. What is the cost of making 25 items?
3. Suppose the maximum cost allowed is \$1500. What are the domain and range of the cost function, $\text{\hspace{0.17em}}C\left(x\right)?$

#### Questions & Answers

what are you up to?
Mark Reply
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
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
sita Reply
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
Pearl Reply
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
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
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 .
Swadesh
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
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
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
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
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply

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