Mixing
$f(x)$ from $f(g(x))$
$begingroup$
Is it always possible to find $f(x)$ if the composite function $h(x) = f(g(x))$ and $g(x)$ are given?
In other words, can there be any cases where, for given $h(x)$, we can not express it in an explicit function of $g(x)$?
functions special-functions function-and-relation-composition
asked Jan 27 at 18:59
JoyJoy
335
$endgroup$
add a comment |
$begingroup$
Is it always possible to find $f(x)$ if the composite function $h(x) = f(g(x))$ and $g(x)$ are given?
In other words, can there be any cases where, for given $h(x)$, we can not express it in an explicit function of $g(x)$?
functions special-functions function-and-relation-composition
asked Jan 27 at 18:59
JoyJoy
335
$endgroup$
$begingroup$
I believe we can find candidates but not the exact function
$endgroup$
– iggykimi
Jan 27 at 19:07
$begingroup$
h(x)=x, g(x)=x^2.. you can't find the negative numbers there
$endgroup$
– Shaq
Jan 27 at 19:09
add a comment |
$begingroup$
Is it always possible to find $f(x)$ if the composite function $h(x) = f(g(x))$ and $g(x)$ are given?
In other words, can there be any cases where, for given $h(x)$, we can not express it in an explicit function of $g(x)$?
functions special-functions function-and-relation-composition
asked Jan 27 at 18:59
JoyJoy
335
$endgroup$
Is it always possible to find $f(x)$ if the composite function $h(x) = f(g(x))$ and $g(x)$ are given?
In other words, can there be any cases where, for given $h(x)$, we can not express it in an explicit function of $g(x)$?
functions special-functions function-and-relation-composition
functions special-functions function-and-relation-composition
asked Jan 27 at 18:59
JoyJoy
335
asked Jan 27 at 18:59
JoyJoy
335
asked Jan 27 at 18:59
JoyJoy
335
asked Jan 27 at 18:59
JoyJoy
335
asked Jan 27 at 18:59
JoyJoy
335
335
$begingroup$
I believe we can find candidates but not the exact function
$endgroup$
– iggykimi
Jan 27 at 19:07
$begingroup$
h(x)=x, g(x)=x^2.. you can't find the negative numbers there
$endgroup$
– Shaq
Jan 27 at 19:09
add a comment |
$begingroup$
I believe we can find candidates but not the exact function
$endgroup$
– iggykimi
Jan 27 at 19:07
$begingroup$
h(x)=x, g(x)=x^2.. you can't find the negative numbers there
$endgroup$
– Shaq
Jan 27 at 19:09
$begingroup$
I believe we can find candidates but not the exact function
$endgroup$
– iggykimi
Jan 27 at 19:07
$begingroup$
I believe we can find candidates but not the exact function
$endgroup$
– iggykimi
Jan 27 at 19:07
$begingroup$
h(x)=x, g(x)=x^2.. you can't find the negative numbers there
$endgroup$
– Shaq
Jan 27 at 19:09
$begingroup$
h(x)=x, g(x)=x^2.. you can't find the negative numbers there
$endgroup$
– Shaq
Jan 27 at 19:09
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Note that if $g(x)$ is invertible then
$$f(x) = h(g^{-1}(x)).$$
What happens if $g(x)$ is not invertible. Consider, e.g., $f(x)=x$ and $g(x) = 1$.
answered Jan 27 at 19:07
Math LoverMath Lover
14.1k31437
$endgroup$
add a comment |
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1 Answer
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oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Note that if $g(x)$ is invertible then
$$f(x) = h(g^{-1}(x)).$$
What happens if $g(x)$ is not invertible. Consider, e.g., $f(x)=x$ and $g(x) = 1$.
answered Jan 27 at 19:07
Math LoverMath Lover
14.1k31437
$endgroup$
add a comment |
$begingroup$
Note that if $g(x)$ is invertible then
$$f(x) = h(g^{-1}(x)).$$
What happens if $g(x)$ is not invertible. Consider, e.g., $f(x)=x$ and $g(x) = 1$.
answered Jan 27 at 19:07
Math LoverMath Lover
14.1k31437
$endgroup$
add a comment |
$begingroup$
Note that if $g(x)$ is invertible then
$$f(x) = h(g^{-1}(x)).$$
What happens if $g(x)$ is not invertible. Consider, e.g., $f(x)=x$ and $g(x) = 1$.
answered Jan 27 at 19:07
Math LoverMath Lover
14.1k31437
$endgroup$
Note that if $g(x)$ is invertible then
$$f(x) = h(g^{-1}(x)).$$
What happens if $g(x)$ is not invertible. Consider, e.g., $f(x)=x$ and $g(x) = 1$.
answered Jan 27 at 19:07
Math LoverMath Lover
14.1k31437
answered Jan 27 at 19:07
Math LoverMath Lover
14.1k31437
answered Jan 27 at 19:07
Math LoverMath Lover
14.1k31437
answered Jan 27 at 19:07
Math LoverMath Lover
14.1k31437
14.1k31437
add a comment |
add a comment |
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$begingroup$
I believe we can find candidates but not the exact function
$endgroup$
– iggykimi
Jan 27 at 19:07
$begingroup$
h(x)=x, g(x)=x^2.. you can't find the negative numbers there
$endgroup$
– Shaq
Jan 27 at 19:09