Prove a function is constant function with the given conditions.
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If $f(4-x)=f (4+x)$ and $f(2-x)=f(2+x)$, prove that $f$ is constant function.
I tried to solve by assuming x and y are two variable then if we get $f (x)=f (y)$ then $f$ is constant function, but can't complete the problem. Someone please help me.
functional-equations
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add a comment |
$begingroup$
If $f(4-x)=f (4+x)$ and $f(2-x)=f(2+x)$, prove that $f$ is constant function.
I tried to solve by assuming x and y are two variable then if we get $f (x)=f (y)$ then $f$ is constant function, but can't complete the problem. Someone please help me.
functional-equations
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Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
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– José Carlos Santos
Oct 27 '18 at 9:21
add a comment |
$begingroup$
If $f(4-x)=f (4+x)$ and $f(2-x)=f(2+x)$, prove that $f$ is constant function.
I tried to solve by assuming x and y are two variable then if we get $f (x)=f (y)$ then $f$ is constant function, but can't complete the problem. Someone please help me.
functional-equations
$endgroup$
If $f(4-x)=f (4+x)$ and $f(2-x)=f(2+x)$, prove that $f$ is constant function.
I tried to solve by assuming x and y are two variable then if we get $f (x)=f (y)$ then $f$ is constant function, but can't complete the problem. Someone please help me.
functional-equations
functional-equations
edited Jan 6 at 2:38
amWhy
1
1
asked Oct 27 '18 at 9:13
Pramit BanerjeePramit Banerjee
11
11
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Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
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– José Carlos Santos
Oct 27 '18 at 9:21
add a comment |
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Oct 27 '18 at 9:21
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Oct 27 '18 at 9:21
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
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– José Carlos Santos
Oct 27 '18 at 9:21
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
$f$ is not necessarily constant. Here is a counter-example:
$$
f(x) =
begin{cases}
0, & text{if $xinBbb Q$} \
1, & text{if $xnotinBbb Q$}
end{cases}$$
And another one, continuous this time:
$$f(x)=cospi x$$
$endgroup$
add a comment |
$begingroup$
The two equations $,f(4-x)=f(4+x),$ and $,f(2-x)=f(2+x),$ require that the function is invariant with respect to two reflections and hence a translation by $4$. The simple calculation is $, x mapsto 8-x ,$ in the first reflection and then $ 8-x mapsto x-4 ,$ after the second reflection yielding a translation by $4.$ Assuming that the function is defined for all real numbers, and that no other conditions apply, then you can let the function be arbitrarily defined on the interval $,[0,2].,$
The function must be an even function with a period of $4$. That is, $,f(x)=f(-x),$ and $,f(x)=f(x+4),$ for all real $,x.,$ This determines the function for all real numbers and that it satisfies $,f(2 n-x)=f(2 n+x)=f(x),$ and $, f(4 n + x) = f(x) ,$ for all integer $,n,$ and all real $,x.,$
For a simple example, consider $, f(4n+x) := |x|,$ for all $,|x|le 2,$ and integer $,n.,$ This is a sawtooth function with fundamental period $,4.,$
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That should read, "and hence a translation by 4." For instance, $f(x)=cosfrac{pi}{2}x$ satisfies the conditions.
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– TonyK
Oct 27 '18 at 10:37
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A reflection about $x=2$, followed by a reflection about $x=4$, amounts to a translation by $4$, not $2$.
$endgroup$
– TonyK
Oct 27 '18 at 10:53
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@TonyK Thanks for that correction. I should be more careful.
$endgroup$
– Somos
Oct 27 '18 at 11:01
add a comment |
$begingroup$
Without tellings what are the domain and the codomain of $f$, nobody can answer that. However, if the domain is, say, $mathbb Q$, then the statement is false. Consider, for instance,$$begin{array}{rccc}fcolon&mathbb Q&longrightarrow&mathbb Z\&q&mapsto&begin{cases}1&text{ if }qinmathbb Z\0&text{ otherwise.}end{cases}end{array}$$
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$f$ is not necessarily constant. Here is a counter-example:
$$
f(x) =
begin{cases}
0, & text{if $xinBbb Q$} \
1, & text{if $xnotinBbb Q$}
end{cases}$$
And another one, continuous this time:
$$f(x)=cospi x$$
$endgroup$
add a comment |
$begingroup$
$f$ is not necessarily constant. Here is a counter-example:
$$
f(x) =
begin{cases}
0, & text{if $xinBbb Q$} \
1, & text{if $xnotinBbb Q$}
end{cases}$$
And another one, continuous this time:
$$f(x)=cospi x$$
$endgroup$
add a comment |
$begingroup$
$f$ is not necessarily constant. Here is a counter-example:
$$
f(x) =
begin{cases}
0, & text{if $xinBbb Q$} \
1, & text{if $xnotinBbb Q$}
end{cases}$$
And another one, continuous this time:
$$f(x)=cospi x$$
$endgroup$
$f$ is not necessarily constant. Here is a counter-example:
$$
f(x) =
begin{cases}
0, & text{if $xinBbb Q$} \
1, & text{if $xnotinBbb Q$}
end{cases}$$
And another one, continuous this time:
$$f(x)=cospi x$$
answered Oct 27 '18 at 9:23
TonyKTonyK
42.2k355134
42.2k355134
add a comment |
add a comment |
$begingroup$
The two equations $,f(4-x)=f(4+x),$ and $,f(2-x)=f(2+x),$ require that the function is invariant with respect to two reflections and hence a translation by $4$. The simple calculation is $, x mapsto 8-x ,$ in the first reflection and then $ 8-x mapsto x-4 ,$ after the second reflection yielding a translation by $4.$ Assuming that the function is defined for all real numbers, and that no other conditions apply, then you can let the function be arbitrarily defined on the interval $,[0,2].,$
The function must be an even function with a period of $4$. That is, $,f(x)=f(-x),$ and $,f(x)=f(x+4),$ for all real $,x.,$ This determines the function for all real numbers and that it satisfies $,f(2 n-x)=f(2 n+x)=f(x),$ and $, f(4 n + x) = f(x) ,$ for all integer $,n,$ and all real $,x.,$
For a simple example, consider $, f(4n+x) := |x|,$ for all $,|x|le 2,$ and integer $,n.,$ This is a sawtooth function with fundamental period $,4.,$
$endgroup$
$begingroup$
That should read, "and hence a translation by 4." For instance, $f(x)=cosfrac{pi}{2}x$ satisfies the conditions.
$endgroup$
– TonyK
Oct 27 '18 at 10:37
$begingroup$
A reflection about $x=2$, followed by a reflection about $x=4$, amounts to a translation by $4$, not $2$.
$endgroup$
– TonyK
Oct 27 '18 at 10:53
$begingroup$
@TonyK Thanks for that correction. I should be more careful.
$endgroup$
– Somos
Oct 27 '18 at 11:01
add a comment |
$begingroup$
The two equations $,f(4-x)=f(4+x),$ and $,f(2-x)=f(2+x),$ require that the function is invariant with respect to two reflections and hence a translation by $4$. The simple calculation is $, x mapsto 8-x ,$ in the first reflection and then $ 8-x mapsto x-4 ,$ after the second reflection yielding a translation by $4.$ Assuming that the function is defined for all real numbers, and that no other conditions apply, then you can let the function be arbitrarily defined on the interval $,[0,2].,$
The function must be an even function with a period of $4$. That is, $,f(x)=f(-x),$ and $,f(x)=f(x+4),$ for all real $,x.,$ This determines the function for all real numbers and that it satisfies $,f(2 n-x)=f(2 n+x)=f(x),$ and $, f(4 n + x) = f(x) ,$ for all integer $,n,$ and all real $,x.,$
For a simple example, consider $, f(4n+x) := |x|,$ for all $,|x|le 2,$ and integer $,n.,$ This is a sawtooth function with fundamental period $,4.,$
$endgroup$
$begingroup$
That should read, "and hence a translation by 4." For instance, $f(x)=cosfrac{pi}{2}x$ satisfies the conditions.
$endgroup$
– TonyK
Oct 27 '18 at 10:37
$begingroup$
A reflection about $x=2$, followed by a reflection about $x=4$, amounts to a translation by $4$, not $2$.
$endgroup$
– TonyK
Oct 27 '18 at 10:53
$begingroup$
@TonyK Thanks for that correction. I should be more careful.
$endgroup$
– Somos
Oct 27 '18 at 11:01
add a comment |
$begingroup$
The two equations $,f(4-x)=f(4+x),$ and $,f(2-x)=f(2+x),$ require that the function is invariant with respect to two reflections and hence a translation by $4$. The simple calculation is $, x mapsto 8-x ,$ in the first reflection and then $ 8-x mapsto x-4 ,$ after the second reflection yielding a translation by $4.$ Assuming that the function is defined for all real numbers, and that no other conditions apply, then you can let the function be arbitrarily defined on the interval $,[0,2].,$
The function must be an even function with a period of $4$. That is, $,f(x)=f(-x),$ and $,f(x)=f(x+4),$ for all real $,x.,$ This determines the function for all real numbers and that it satisfies $,f(2 n-x)=f(2 n+x)=f(x),$ and $, f(4 n + x) = f(x) ,$ for all integer $,n,$ and all real $,x.,$
For a simple example, consider $, f(4n+x) := |x|,$ for all $,|x|le 2,$ and integer $,n.,$ This is a sawtooth function with fundamental period $,4.,$
$endgroup$
The two equations $,f(4-x)=f(4+x),$ and $,f(2-x)=f(2+x),$ require that the function is invariant with respect to two reflections and hence a translation by $4$. The simple calculation is $, x mapsto 8-x ,$ in the first reflection and then $ 8-x mapsto x-4 ,$ after the second reflection yielding a translation by $4.$ Assuming that the function is defined for all real numbers, and that no other conditions apply, then you can let the function be arbitrarily defined on the interval $,[0,2].,$
The function must be an even function with a period of $4$. That is, $,f(x)=f(-x),$ and $,f(x)=f(x+4),$ for all real $,x.,$ This determines the function for all real numbers and that it satisfies $,f(2 n-x)=f(2 n+x)=f(x),$ and $, f(4 n + x) = f(x) ,$ for all integer $,n,$ and all real $,x.,$
For a simple example, consider $, f(4n+x) := |x|,$ for all $,|x|le 2,$ and integer $,n.,$ This is a sawtooth function with fundamental period $,4.,$
edited Oct 27 '18 at 11:05
answered Oct 27 '18 at 10:08
SomosSomos
13.1k11034
13.1k11034
$begingroup$
That should read, "and hence a translation by 4." For instance, $f(x)=cosfrac{pi}{2}x$ satisfies the conditions.
$endgroup$
– TonyK
Oct 27 '18 at 10:37
$begingroup$
A reflection about $x=2$, followed by a reflection about $x=4$, amounts to a translation by $4$, not $2$.
$endgroup$
– TonyK
Oct 27 '18 at 10:53
$begingroup$
@TonyK Thanks for that correction. I should be more careful.
$endgroup$
– Somos
Oct 27 '18 at 11:01
add a comment |
$begingroup$
That should read, "and hence a translation by 4." For instance, $f(x)=cosfrac{pi}{2}x$ satisfies the conditions.
$endgroup$
– TonyK
Oct 27 '18 at 10:37
$begingroup$
A reflection about $x=2$, followed by a reflection about $x=4$, amounts to a translation by $4$, not $2$.
$endgroup$
– TonyK
Oct 27 '18 at 10:53
$begingroup$
@TonyK Thanks for that correction. I should be more careful.
$endgroup$
– Somos
Oct 27 '18 at 11:01
$begingroup$
That should read, "and hence a translation by 4." For instance, $f(x)=cosfrac{pi}{2}x$ satisfies the conditions.
$endgroup$
– TonyK
Oct 27 '18 at 10:37
$begingroup$
That should read, "and hence a translation by 4." For instance, $f(x)=cosfrac{pi}{2}x$ satisfies the conditions.
$endgroup$
– TonyK
Oct 27 '18 at 10:37
$begingroup$
A reflection about $x=2$, followed by a reflection about $x=4$, amounts to a translation by $4$, not $2$.
$endgroup$
– TonyK
Oct 27 '18 at 10:53
$begingroup$
A reflection about $x=2$, followed by a reflection about $x=4$, amounts to a translation by $4$, not $2$.
$endgroup$
– TonyK
Oct 27 '18 at 10:53
$begingroup$
@TonyK Thanks for that correction. I should be more careful.
$endgroup$
– Somos
Oct 27 '18 at 11:01
$begingroup$
@TonyK Thanks for that correction. I should be more careful.
$endgroup$
– Somos
Oct 27 '18 at 11:01
add a comment |
$begingroup$
Without tellings what are the domain and the codomain of $f$, nobody can answer that. However, if the domain is, say, $mathbb Q$, then the statement is false. Consider, for instance,$$begin{array}{rccc}fcolon&mathbb Q&longrightarrow&mathbb Z\&q&mapsto&begin{cases}1&text{ if }qinmathbb Z\0&text{ otherwise.}end{cases}end{array}$$
$endgroup$
add a comment |
$begingroup$
Without tellings what are the domain and the codomain of $f$, nobody can answer that. However, if the domain is, say, $mathbb Q$, then the statement is false. Consider, for instance,$$begin{array}{rccc}fcolon&mathbb Q&longrightarrow&mathbb Z\&q&mapsto&begin{cases}1&text{ if }qinmathbb Z\0&text{ otherwise.}end{cases}end{array}$$
$endgroup$
add a comment |
$begingroup$
Without tellings what are the domain and the codomain of $f$, nobody can answer that. However, if the domain is, say, $mathbb Q$, then the statement is false. Consider, for instance,$$begin{array}{rccc}fcolon&mathbb Q&longrightarrow&mathbb Z\&q&mapsto&begin{cases}1&text{ if }qinmathbb Z\0&text{ otherwise.}end{cases}end{array}$$
$endgroup$
Without tellings what are the domain and the codomain of $f$, nobody can answer that. However, if the domain is, say, $mathbb Q$, then the statement is false. Consider, for instance,$$begin{array}{rccc}fcolon&mathbb Q&longrightarrow&mathbb Z\&q&mapsto&begin{cases}1&text{ if }qinmathbb Z\0&text{ otherwise.}end{cases}end{array}$$
answered Oct 27 '18 at 9:25
José Carlos SantosJosé Carlos Santos
154k22123226
154k22123226
add a comment |
add a comment |
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$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Oct 27 '18 at 9:21