Question

Graph the functions and describe the domain and range of each function. f(x)=-square root x+1

Answer 1

is it
\[ f(x)=-\sqrt{x+1} \]
?

Answer 2

yes!

Answer 3

Ok for a square root to have sense, what is inside has to be
(i) positive or zero, or
(ii) negative

Answer 4

ok..

Answer 5

it was a question! choose (i) or (ii)

Answer 6

positive

Answer 7

positive or zero.
remember
\[ \sqrt{0}=0 \]
Then you have to solve the inequality
\[ x+1\geq0 \]

Answer 8

0?

Answer 9

it that zero or Ohhh?

Answer 10

zero

Answer 11

it is an inequality!
\[ x+1\geq0\qquad\Rightarrow\qquad x\geq-1 \]
right?

Answer 12

yes..

Answer 13

so what is the domain of the function?

Answer 14

1?

Answer 15

the expression x>=-1 corresponds to an interval
what interval?

Answer 16

-1?

Answer 17

do you know any of these
\[ [a,b]\qquad (a.b)\qquad [a,b)\qquad (a.b]\qquad
[a,\infty)\qquad (-\infty,b] \]
?

Answer 18

i recognize them.

Answer 19

do you know it definitions?

Answer 20

infinite

Answer 21

OK
\[ [a,b]=\{x\in\mathbb{R}:a\leq x\leq b\} \]
\[ (a,b)=\{x\in\mathbb{R}:a<x< b\} \]
\[ [a,\infty)=\{x\in\mathbb{R}:x\geq a\} \]

Answer 22

have you ever seen this?

Answer 23

yes..but i do NOT know how to identify it. can you show me how to do this first problem so i can maybe try the other ones..

Answer 24

You have to check the right sides of the sets i gave you (what's after the colon).
the expression
\[ x\geq-1 \]
corresponds to which of the three examples i gave u?

Answer 25

c

Answer 26

Yes!
so the domain of the function is
\[ D_f=[-1,\infty) \]
right?

Answer 27

yes

Answer 28

did you understand ?

Answer 29

a little?

Answer 30

ok we'll get to it later.
now for the range of the function first solve for x the equation
\[ y=f(x)=-\sqrt{x+1} \]

Answer 31

post what you get

Answer 32

WHAT WOULD I PUT FOR X?!

Answer 33

nothing! just x!

Answer 34

ok..now what.

Answer 35

what did you get?

Answer 36

if X is nothing then what am i solving for! +1?

Answer 37

let me do it! you just write x!
\[ \LARGE y=f(x)=-\sqrt{x+1} \]
\[ \LARGE -y=\sqrt{x+1} \]
\[ \LARGE y^2=(-y)^2=x+1 \]
\[ \LARGE x=y^2-1 \]
OK?

Answer 38

..yeah what is x

Answer 39

just x!

Answer 40

you keep saying write x..but x isnt anything

Answer 41

x and y are variables, y depends on x, and x is independent
right?

Answer 42

ya

Answer 43

when you solve the equation i solved you are finding the inverse function (or relation if f is not inyective)

Answer 44

now in the expression
\[ \LARGE -y=\sqrt{x+1} \]
we have a restriction
\[ \LARGE \sqrt{\text{?}}\geq0 \]
so this means that
\[ \LARGE -y\geq0\Rightarrow y\leq0 \]
this gives us the range
\[ \LARGE R_f=(-\infty,0] \]

Answer 45

are there? tired or bored?

Answer 46

sorry..i stepped out for a while

Answer 47

no problem
did you understand?