Domain of functions involving arcsine or arccosine
up vote
2
down vote
favorite
I have two rather simple functions. I don't know how to find that the dominator is different than 0. Find the domain of functions:
Function 1:
$f(x) = frac{4-x}{arcsinfrac{x}{4}}$
Assumption 1:
$-1leq arcsinfrac{x}{4} leq 1$
$-1leq frac{x}{4} leq 1$
$-4leq x leq 4$
$xin<4;4>$
Assumption 2:
$arcsinfrac{x}{4} neq 0$
Here, I have no idea how to proceed further with assumption 2.
Function 2:
$f(x) = frac{sqrt{2x-1}}{2+arccosfrac{x+1}{4}}$
Assumption 1:
$sqrt{2x-1} geq 0$
$2x geq 1$
$x geq frac{1}{2}$
Assumption 2:
$ -1 leq arccos frac{x+1}{4} leq 1 $
$ -1 leq frac{x+1}{4} leq 1 $
$ -4 leq x + 1 leq 4 $
$ -5 leq x leq 3 $
Assumption 3:
$ 2 + arccosfrac{x+1}{4} neq 0 $
Here once again, no idea how to proceed further.
Would anyone be able to help me understand these examples, how to find that the denominator is not equal to $0$, when there's arccos or arcsin in the denominator? Thanks! I do hope my tags are okay...
functions trigonometry
add a comment |
up vote
2
down vote
favorite
I have two rather simple functions. I don't know how to find that the dominator is different than 0. Find the domain of functions:
Function 1:
$f(x) = frac{4-x}{arcsinfrac{x}{4}}$
Assumption 1:
$-1leq arcsinfrac{x}{4} leq 1$
$-1leq frac{x}{4} leq 1$
$-4leq x leq 4$
$xin<4;4>$
Assumption 2:
$arcsinfrac{x}{4} neq 0$
Here, I have no idea how to proceed further with assumption 2.
Function 2:
$f(x) = frac{sqrt{2x-1}}{2+arccosfrac{x+1}{4}}$
Assumption 1:
$sqrt{2x-1} geq 0$
$2x geq 1$
$x geq frac{1}{2}$
Assumption 2:
$ -1 leq arccos frac{x+1}{4} leq 1 $
$ -1 leq frac{x+1}{4} leq 1 $
$ -4 leq x + 1 leq 4 $
$ -5 leq x leq 3 $
Assumption 3:
$ 2 + arccosfrac{x+1}{4} neq 0 $
Here once again, no idea how to proceed further.
Would anyone be able to help me understand these examples, how to find that the denominator is not equal to $0$, when there's arccos or arcsin in the denominator? Thanks! I do hope my tags are okay...
functions trigonometry
Use en.wikipedia.org/wiki/…
– lab bhattacharjee
Nov 21 at 13:56
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I have two rather simple functions. I don't know how to find that the dominator is different than 0. Find the domain of functions:
Function 1:
$f(x) = frac{4-x}{arcsinfrac{x}{4}}$
Assumption 1:
$-1leq arcsinfrac{x}{4} leq 1$
$-1leq frac{x}{4} leq 1$
$-4leq x leq 4$
$xin<4;4>$
Assumption 2:
$arcsinfrac{x}{4} neq 0$
Here, I have no idea how to proceed further with assumption 2.
Function 2:
$f(x) = frac{sqrt{2x-1}}{2+arccosfrac{x+1}{4}}$
Assumption 1:
$sqrt{2x-1} geq 0$
$2x geq 1$
$x geq frac{1}{2}$
Assumption 2:
$ -1 leq arccos frac{x+1}{4} leq 1 $
$ -1 leq frac{x+1}{4} leq 1 $
$ -4 leq x + 1 leq 4 $
$ -5 leq x leq 3 $
Assumption 3:
$ 2 + arccosfrac{x+1}{4} neq 0 $
Here once again, no idea how to proceed further.
Would anyone be able to help me understand these examples, how to find that the denominator is not equal to $0$, when there's arccos or arcsin in the denominator? Thanks! I do hope my tags are okay...
functions trigonometry
I have two rather simple functions. I don't know how to find that the dominator is different than 0. Find the domain of functions:
Function 1:
$f(x) = frac{4-x}{arcsinfrac{x}{4}}$
Assumption 1:
$-1leq arcsinfrac{x}{4} leq 1$
$-1leq frac{x}{4} leq 1$
$-4leq x leq 4$
$xin<4;4>$
Assumption 2:
$arcsinfrac{x}{4} neq 0$
Here, I have no idea how to proceed further with assumption 2.
Function 2:
$f(x) = frac{sqrt{2x-1}}{2+arccosfrac{x+1}{4}}$
Assumption 1:
$sqrt{2x-1} geq 0$
$2x geq 1$
$x geq frac{1}{2}$
Assumption 2:
$ -1 leq arccos frac{x+1}{4} leq 1 $
$ -1 leq frac{x+1}{4} leq 1 $
$ -4 leq x + 1 leq 4 $
$ -5 leq x leq 3 $
Assumption 3:
$ 2 + arccosfrac{x+1}{4} neq 0 $
Here once again, no idea how to proceed further.
Would anyone be able to help me understand these examples, how to find that the denominator is not equal to $0$, when there's arccos or arcsin in the denominator? Thanks! I do hope my tags are okay...
functions trigonometry
functions trigonometry
edited Nov 21 at 15:10
N. F. Taussig
43.1k93254
43.1k93254
asked Nov 21 at 13:54
weno
657
657
Use en.wikipedia.org/wiki/…
– lab bhattacharjee
Nov 21 at 13:56
add a comment |
Use en.wikipedia.org/wiki/…
– lab bhattacharjee
Nov 21 at 13:56
Use en.wikipedia.org/wiki/…
– lab bhattacharjee
Nov 21 at 13:56
Use en.wikipedia.org/wiki/…
– lab bhattacharjee
Nov 21 at 13:56
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
For the first function, the fraction is defined when $arcsin frac{x}{4} ne 0 implies frac{x}{4} ne sin 0=0 implies xne 0$
So the domain of the first function is $[-4,4]-{0}$
For the second function $$2+arccos frac{x+1}{4} ne 0$$
$$implies arccos frac{x+1}{4} ne -2$$
Apply cos both sides,
$$implies frac{x+1}{4} ne cos(-2)$$
$$implies x ne 4cos(-2) -1$$
2
But since $arcsin$ is defined only on the closed interval $[-1,1]$, the denominator in #1 is defined only on the closed interval $[-4,4]$. Outside of that interval, the whole is undefined.
– Lubin
Nov 21 at 15:03
The range of $arccos x$ is $[0, pi]$. Hence, the range of $2 + arccosleft(frac{x + 1}{4}right)$ is $[2, 2 + pi]$.
– N. F. Taussig
Nov 22 at 13:07
1
Type$sin x$
,$cos x$
,$tan x$
,$csc x$
,$sec x$
,$cot x$
,$arcsin x$
,$arccos x$
,$arctan x$
to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, $cot x$, $arcsin x$, $arccos x$, and $arctan x$, respectively.
– N. F. Taussig
Nov 22 at 13:09
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
For the first function, the fraction is defined when $arcsin frac{x}{4} ne 0 implies frac{x}{4} ne sin 0=0 implies xne 0$
So the domain of the first function is $[-4,4]-{0}$
For the second function $$2+arccos frac{x+1}{4} ne 0$$
$$implies arccos frac{x+1}{4} ne -2$$
Apply cos both sides,
$$implies frac{x+1}{4} ne cos(-2)$$
$$implies x ne 4cos(-2) -1$$
2
But since $arcsin$ is defined only on the closed interval $[-1,1]$, the denominator in #1 is defined only on the closed interval $[-4,4]$. Outside of that interval, the whole is undefined.
– Lubin
Nov 21 at 15:03
The range of $arccos x$ is $[0, pi]$. Hence, the range of $2 + arccosleft(frac{x + 1}{4}right)$ is $[2, 2 + pi]$.
– N. F. Taussig
Nov 22 at 13:07
1
Type$sin x$
,$cos x$
,$tan x$
,$csc x$
,$sec x$
,$cot x$
,$arcsin x$
,$arccos x$
,$arctan x$
to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, $cot x$, $arcsin x$, $arccos x$, and $arctan x$, respectively.
– N. F. Taussig
Nov 22 at 13:09
add a comment |
up vote
0
down vote
For the first function, the fraction is defined when $arcsin frac{x}{4} ne 0 implies frac{x}{4} ne sin 0=0 implies xne 0$
So the domain of the first function is $[-4,4]-{0}$
For the second function $$2+arccos frac{x+1}{4} ne 0$$
$$implies arccos frac{x+1}{4} ne -2$$
Apply cos both sides,
$$implies frac{x+1}{4} ne cos(-2)$$
$$implies x ne 4cos(-2) -1$$
2
But since $arcsin$ is defined only on the closed interval $[-1,1]$, the denominator in #1 is defined only on the closed interval $[-4,4]$. Outside of that interval, the whole is undefined.
– Lubin
Nov 21 at 15:03
The range of $arccos x$ is $[0, pi]$. Hence, the range of $2 + arccosleft(frac{x + 1}{4}right)$ is $[2, 2 + pi]$.
– N. F. Taussig
Nov 22 at 13:07
1
Type$sin x$
,$cos x$
,$tan x$
,$csc x$
,$sec x$
,$cot x$
,$arcsin x$
,$arccos x$
,$arctan x$
to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, $cot x$, $arcsin x$, $arccos x$, and $arctan x$, respectively.
– N. F. Taussig
Nov 22 at 13:09
add a comment |
up vote
0
down vote
up vote
0
down vote
For the first function, the fraction is defined when $arcsin frac{x}{4} ne 0 implies frac{x}{4} ne sin 0=0 implies xne 0$
So the domain of the first function is $[-4,4]-{0}$
For the second function $$2+arccos frac{x+1}{4} ne 0$$
$$implies arccos frac{x+1}{4} ne -2$$
Apply cos both sides,
$$implies frac{x+1}{4} ne cos(-2)$$
$$implies x ne 4cos(-2) -1$$
For the first function, the fraction is defined when $arcsin frac{x}{4} ne 0 implies frac{x}{4} ne sin 0=0 implies xne 0$
So the domain of the first function is $[-4,4]-{0}$
For the second function $$2+arccos frac{x+1}{4} ne 0$$
$$implies arccos frac{x+1}{4} ne -2$$
Apply cos both sides,
$$implies frac{x+1}{4} ne cos(-2)$$
$$implies x ne 4cos(-2) -1$$
edited Nov 23 at 6:44
answered Nov 21 at 14:11
Fareed AF
41211
41211
2
But since $arcsin$ is defined only on the closed interval $[-1,1]$, the denominator in #1 is defined only on the closed interval $[-4,4]$. Outside of that interval, the whole is undefined.
– Lubin
Nov 21 at 15:03
The range of $arccos x$ is $[0, pi]$. Hence, the range of $2 + arccosleft(frac{x + 1}{4}right)$ is $[2, 2 + pi]$.
– N. F. Taussig
Nov 22 at 13:07
1
Type$sin x$
,$cos x$
,$tan x$
,$csc x$
,$sec x$
,$cot x$
,$arcsin x$
,$arccos x$
,$arctan x$
to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, $cot x$, $arcsin x$, $arccos x$, and $arctan x$, respectively.
– N. F. Taussig
Nov 22 at 13:09
add a comment |
2
But since $arcsin$ is defined only on the closed interval $[-1,1]$, the denominator in #1 is defined only on the closed interval $[-4,4]$. Outside of that interval, the whole is undefined.
– Lubin
Nov 21 at 15:03
The range of $arccos x$ is $[0, pi]$. Hence, the range of $2 + arccosleft(frac{x + 1}{4}right)$ is $[2, 2 + pi]$.
– N. F. Taussig
Nov 22 at 13:07
1
Type$sin x$
,$cos x$
,$tan x$
,$csc x$
,$sec x$
,$cot x$
,$arcsin x$
,$arccos x$
,$arctan x$
to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, $cot x$, $arcsin x$, $arccos x$, and $arctan x$, respectively.
– N. F. Taussig
Nov 22 at 13:09
2
2
But since $arcsin$ is defined only on the closed interval $[-1,1]$, the denominator in #1 is defined only on the closed interval $[-4,4]$. Outside of that interval, the whole is undefined.
– Lubin
Nov 21 at 15:03
But since $arcsin$ is defined only on the closed interval $[-1,1]$, the denominator in #1 is defined only on the closed interval $[-4,4]$. Outside of that interval, the whole is undefined.
– Lubin
Nov 21 at 15:03
The range of $arccos x$ is $[0, pi]$. Hence, the range of $2 + arccosleft(frac{x + 1}{4}right)$ is $[2, 2 + pi]$.
– N. F. Taussig
Nov 22 at 13:07
The range of $arccos x$ is $[0, pi]$. Hence, the range of $2 + arccosleft(frac{x + 1}{4}right)$ is $[2, 2 + pi]$.
– N. F. Taussig
Nov 22 at 13:07
1
1
Type
$sin x$
, $cos x$
, $tan x$
, $csc x$
, $sec x$
, $cot x$
, $arcsin x$
, $arccos x$
, $arctan x$
to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, $cot x$, $arcsin x$, $arccos x$, and $arctan x$, respectively.– N. F. Taussig
Nov 22 at 13:09
Type
$sin x$
, $cos x$
, $tan x$
, $csc x$
, $sec x$
, $cot x$
, $arcsin x$
, $arccos x$
, $arctan x$
to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, $cot x$, $arcsin x$, $arccos x$, and $arctan x$, respectively.– N. F. Taussig
Nov 22 at 13:09
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3007759%2fdomain-of-functions-involving-arcsine-or-arccosine%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Use en.wikipedia.org/wiki/…
– lab bhattacharjee
Nov 21 at 13:56