Range of Compositions of Functions
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Is there an efficient method to find the range of compositions of functions?
I know this: The domain of the composition of functions is the INTERSECTION of the domain of the INSIDE function and the domain of the RESULTING function.
However, I'm struggling to find the range of compositions of functions. Is the range the INTERSECTION of the range of the OUTSIDE function and the range of the resulting function? I don't think this assumption is correct though. Can someone please help?
For example, let's define two functions: $f(x)= sqrt{x}$ and $g(x)= x+1$. The resulting function, $f(g(x))$ is $sqrt{x+1}$. This resulting function has a domain of greater than or equal to $-1$, and it has a range of greater than or equal to $0$.
In the example, $f(x)$ has a domain of greater than or equal to $0$, $g(x)$ has a domain of all real numbers, $f(x)$ has a range of greater than or equal to $0$, $g(x)$ has a range of all real numbers.
The domain of the composition is the INTERSECTION of the domain of the INSIDE function (in this case, $g(x)$ has a domain of all real numbers), and the domain of the RESULTING function (in this case, $f(g(x))$, or $sqrt{x+1}$, has a domain of greater than or equal to $-1$). Hence, the final domain of the composition is GREATER THAN OR EQUAL TO $-1$.
What is the range??
Thank you!
algebra-precalculus functions
edited Dec 16 '18 at 10:54
N. F. Taussig
44.6k103357
asked Dec 16 '18 at 5:10
Agastya RaoAgastya Rao
11
$endgroup$
add a comment |
$begingroup$
Is there an efficient method to find the range of compositions of functions?
I know this: The domain of the composition of functions is the INTERSECTION of the domain of the INSIDE function and the domain of the RESULTING function.
However, I'm struggling to find the range of compositions of functions. Is the range the INTERSECTION of the range of the OUTSIDE function and the range of the resulting function? I don't think this assumption is correct though. Can someone please help?
For example, let's define two functions: $f(x)= sqrt{x}$ and $g(x)= x+1$. The resulting function, $f(g(x))$ is $sqrt{x+1}$. This resulting function has a domain of greater than or equal to $-1$, and it has a range of greater than or equal to $0$.
In the example, $f(x)$ has a domain of greater than or equal to $0$, $g(x)$ has a domain of all real numbers, $f(x)$ has a range of greater than or equal to $0$, $g(x)$ has a range of all real numbers.
The domain of the composition is the INTERSECTION of the domain of the INSIDE function (in this case, $g(x)$ has a domain of all real numbers), and the domain of the RESULTING function (in this case, $f(g(x))$, or $sqrt{x+1}$, has a domain of greater than or equal to $-1$). Hence, the final domain of the composition is GREATER THAN OR EQUAL TO $-1$.
What is the range??
Thank you!
algebra-precalculus functions
edited Dec 16 '18 at 10:54
N. F. Taussig
44.6k103357
asked Dec 16 '18 at 5:10
Agastya RaoAgastya Rao
11
$endgroup$
$begingroup$
What is meant by the "RESULTING function"? [If it's the composition no need to intersect.]
$endgroup$
– coffeemath
Dec 16 '18 at 5:18
$begingroup$
I have edited my post, thanks.
$endgroup$
– Agastya Rao
Dec 16 '18 at 5:42
$begingroup$
Welcome to MathSE. This tutorial explains how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 16 '18 at 10:52
add a comment |
$begingroup$
Is there an efficient method to find the range of compositions of functions?
I know this: The domain of the composition of functions is the INTERSECTION of the domain of the INSIDE function and the domain of the RESULTING function.
However, I'm struggling to find the range of compositions of functions. Is the range the INTERSECTION of the range of the OUTSIDE function and the range of the resulting function? I don't think this assumption is correct though. Can someone please help?
For example, let's define two functions: $f(x)= sqrt{x}$ and $g(x)= x+1$. The resulting function, $f(g(x))$ is $sqrt{x+1}$. This resulting function has a domain of greater than or equal to $-1$, and it has a range of greater than or equal to $0$.
In the example, $f(x)$ has a domain of greater than or equal to $0$, $g(x)$ has a domain of all real numbers, $f(x)$ has a range of greater than or equal to $0$, $g(x)$ has a range of all real numbers.
The domain of the composition is the INTERSECTION of the domain of the INSIDE function (in this case, $g(x)$ has a domain of all real numbers), and the domain of the RESULTING function (in this case, $f(g(x))$, or $sqrt{x+1}$, has a domain of greater than or equal to $-1$). Hence, the final domain of the composition is GREATER THAN OR EQUAL TO $-1$.
What is the range??
Thank you!
algebra-precalculus functions
edited Dec 16 '18 at 10:54
N. F. Taussig
44.6k103357
asked Dec 16 '18 at 5:10
Agastya RaoAgastya Rao
11
$endgroup$
Is there an efficient method to find the range of compositions of functions?
I know this: The domain of the composition of functions is the INTERSECTION of the domain of the INSIDE function and the domain of the RESULTING function.
However, I'm struggling to find the range of compositions of functions. Is the range the INTERSECTION of the range of the OUTSIDE function and the range of the resulting function? I don't think this assumption is correct though. Can someone please help?
For example, let's define two functions: $f(x)= sqrt{x}$ and $g(x)= x+1$. The resulting function, $f(g(x))$ is $sqrt{x+1}$. This resulting function has a domain of greater than or equal to $-1$, and it has a range of greater than or equal to $0$.
In the example, $f(x)$ has a domain of greater than or equal to $0$, $g(x)$ has a domain of all real numbers, $f(x)$ has a range of greater than or equal to $0$, $g(x)$ has a range of all real numbers.
The domain of the composition is the INTERSECTION of the domain of the INSIDE function (in this case, $g(x)$ has a domain of all real numbers), and the domain of the RESULTING function (in this case, $f(g(x))$, or $sqrt{x+1}$, has a domain of greater than or equal to $-1$). Hence, the final domain of the composition is GREATER THAN OR EQUAL TO $-1$.
What is the range??
Thank you!
algebra-precalculus functions
algebra-precalculus functions
edited Dec 16 '18 at 10:54
N. F. Taussig
44.6k103357
asked Dec 16 '18 at 5:10
Agastya RaoAgastya Rao
11
edited Dec 16 '18 at 10:54
N. F. Taussig
44.6k103357
asked Dec 16 '18 at 5:10
Agastya RaoAgastya Rao
11
edited Dec 16 '18 at 10:54
N. F. Taussig
44.6k103357
edited Dec 16 '18 at 10:54
N. F. Taussig
44.6k103357
edited Dec 16 '18 at 10:54
N. F. Taussig
44.6k103357
44.6k103357
asked Dec 16 '18 at 5:10
Agastya RaoAgastya Rao
11
asked Dec 16 '18 at 5:10
Agastya RaoAgastya Rao
11
asked Dec 16 '18 at 5:10
Agastya RaoAgastya Rao
11
11
$begingroup$
What is meant by the "RESULTING function"? [If it's the composition no need to intersect.]
$endgroup$
– coffeemath
Dec 16 '18 at 5:18
$begingroup$
I have edited my post, thanks.
$endgroup$
– Agastya Rao
Dec 16 '18 at 5:42
$begingroup$
Welcome to MathSE. This tutorial explains how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 16 '18 at 10:52
add a comment |
$begingroup$
What is meant by the "RESULTING function"? [If it's the composition no need to intersect.]
$endgroup$
– coffeemath
Dec 16 '18 at 5:18
$begingroup$
I have edited my post, thanks.
$endgroup$
– Agastya Rao
Dec 16 '18 at 5:42
$begingroup$
Welcome to MathSE. This tutorial explains how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 16 '18 at 10:52
$begingroup$
What is meant by the "RESULTING function"? [If it's the composition no need to intersect.]
$endgroup$
– coffeemath
Dec 16 '18 at 5:18
$begingroup$
What is meant by the "RESULTING function"? [If it's the composition no need to intersect.]
$endgroup$
– coffeemath
Dec 16 '18 at 5:18
$begingroup$
I have edited my post, thanks.
$endgroup$
– Agastya Rao
Dec 16 '18 at 5:42
$begingroup$
I have edited my post, thanks.
$endgroup$
– Agastya Rao
Dec 16 '18 at 5:42
$begingroup$
Welcome to MathSE. This tutorial explains how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 16 '18 at 10:52
$begingroup$
Welcome to MathSE. This tutorial explains how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 16 '18 at 10:52
add a comment |
1 Answer
1
active
oldest
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Yes. If f:X -> Y, g:Y -> Z.
Range g $circ$ f is g(f(X)).
Notation. If h function, A subset domain h, then
h(A) = { h(x) : x in A }.
In your example domain f = [0,oo).
Composition f o g is not possible because
range g is not subset of domain f.
Let h be g restriced to a subset A of domain g.
If g(A) subset domain f, ie range h subset domain f,
then the composition f o h is possible and
range f o h = f(h(A)).
Likely you would want the largest A subset domain g
with g(A) subset domain f.
answered Dec 16 '18 at 10:35
William ElliotWilliam Elliot
8,6222720
$endgroup$
$begingroup$
Please read this tutorial on how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 16 '18 at 10:56
add a comment |
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1 Answer
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1 Answer
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oldest
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active
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$begingroup$
Yes. If f:X -> Y, g:Y -> Z.
Range g $circ$ f is g(f(X)).
Notation. If h function, A subset domain h, then
h(A) = { h(x) : x in A }.
In your example domain f = [0,oo).
Composition f o g is not possible because
range g is not subset of domain f.
Let h be g restriced to a subset A of domain g.
If g(A) subset domain f, ie range h subset domain f,
then the composition f o h is possible and
range f o h = f(h(A)).
Likely you would want the largest A subset domain g
with g(A) subset domain f.
answered Dec 16 '18 at 10:35
William ElliotWilliam Elliot
8,6222720
$endgroup$
$begingroup$
Please read this tutorial on how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 16 '18 at 10:56
add a comment |
$begingroup$
Yes. If f:X -> Y, g:Y -> Z.
Range g $circ$ f is g(f(X)).
Notation. If h function, A subset domain h, then
h(A) = { h(x) : x in A }.
In your example domain f = [0,oo).
Composition f o g is not possible because
range g is not subset of domain f.
Let h be g restriced to a subset A of domain g.
If g(A) subset domain f, ie range h subset domain f,
then the composition f o h is possible and
range f o h = f(h(A)).
Likely you would want the largest A subset domain g
with g(A) subset domain f.
answered Dec 16 '18 at 10:35
William ElliotWilliam Elliot
8,6222720
$endgroup$
$begingroup$
Please read this tutorial on how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 16 '18 at 10:56
add a comment |
$begingroup$
Yes. If f:X -> Y, g:Y -> Z.
Range g $circ$ f is g(f(X)).
Notation. If h function, A subset domain h, then
h(A) = { h(x) : x in A }.
In your example domain f = [0,oo).
Composition f o g is not possible because
range g is not subset of domain f.
Let h be g restriced to a subset A of domain g.
If g(A) subset domain f, ie range h subset domain f,
then the composition f o h is possible and
range f o h = f(h(A)).
Likely you would want the largest A subset domain g
with g(A) subset domain f.
answered Dec 16 '18 at 10:35
William ElliotWilliam Elliot
8,6222720
$endgroup$
Yes. If f:X -> Y, g:Y -> Z.
Range g $circ$ f is g(f(X)).
Notation. If h function, A subset domain h, then
h(A) = { h(x) : x in A }.
In your example domain f = [0,oo).
Composition f o g is not possible because
range g is not subset of domain f.
Let h be g restriced to a subset A of domain g.
If g(A) subset domain f, ie range h subset domain f,
then the composition f o h is possible and
range f o h = f(h(A)).
Likely you would want the largest A subset domain g
with g(A) subset domain f.
answered Dec 16 '18 at 10:35
William ElliotWilliam Elliot
8,6222720
answered Dec 16 '18 at 10:35
William ElliotWilliam Elliot
8,6222720
answered Dec 16 '18 at 10:35
William ElliotWilliam Elliot
8,6222720
answered Dec 16 '18 at 10:35
William ElliotWilliam Elliot
8,6222720
8,6222720
$begingroup$
Please read this tutorial on how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 16 '18 at 10:56
add a comment |
$begingroup$
Please read this tutorial on how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 16 '18 at 10:56
$begingroup$
Please read this tutorial on how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 16 '18 at 10:56
$begingroup$
Please read this tutorial on how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 16 '18 at 10:56
add a comment |
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$begingroup$
What is meant by the "RESULTING function"? [If it's the composition no need to intersect.]
$endgroup$
– coffeemath
Dec 16 '18 at 5:18
$begingroup$
I have edited my post, thanks.
$endgroup$
– Agastya Rao
Dec 16 '18 at 5:42
$begingroup$
Welcome to MathSE. This tutorial explains how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 16 '18 at 10:52