Finding an Inverse Function and Composition of Functions? I add for all y ∈ R [duplicate]











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  • Finding an Inverse Function and Composition of Functions?

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The functions of each pair are inverse to each other. For each pair, check that both compositions give the identity function.



$F: mathbb{R} to mathbb{R}$ and $F^{−1}:mathbb{R} to mathbb{R}$ are defined by $F(x)=3x+2$ and $F^{−1}(y)=dfrac{y−2}{3}$, for all $y in Bbb{R}.$



My attempt:



Inverse Function



For each particular but arbitrarily chosen $y in mathbb{R}$, according to the definition of $f^{-1}$, $f^{-1}(y) = dfrac{y-2}{3}$ is a unique real number $x$ such that $f(x) = y$.
begin{align*}
F(x) & = y\
3x + 2 & = y\
x & = frac{y-2}{3}
end{align*}

Therefore, $f^{-1}(y) = frac{y-2}{3}$.



Compositions of Functions.



The functions $g circ f$ and $f circ g$ are defined as follows:
$$(g circ f)(x) = g(f(x)) = g(3x + 2) = 3x + 2$$
for all $x in mathbb{Z}$.










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marked as duplicate by N. F. Taussig, Community Nov 17 at 16:53


This question was marked as an exact duplicate of an existing question.















  • What is $g$ here?
    – drhab
    Nov 17 at 10:06










  • @drhab (g ◦ f )(x) = g( f (x)) for all x ∈ X, where g ◦ f is read “g circle f ” and g( f (x)) is read “g of f of x.” The function g ◦ f is called the composition of f and g.
    – adam sala
    Nov 17 at 10:10










  • I know, but in the last lines of your question $g$ just seems to fall from the sky. Further it is said that $g(3x+2)=3x+2$ which implies that $g$ is the identity. Is that really what you meant to say?
    – drhab
    Nov 17 at 10:13










  • yes @drhab , is the composition of function above correct?
    – adam sala
    Nov 17 at 10:23










  • If $f$ is prescribed by $xmapsto3x+2$ and $g$ is the identity function then $gcirc f$ is indeed prescribed by $xmapsto 3x+2$. Actually then $gcirc f=f$. But I do not see any functionality of this in this context.
    – drhab
    Nov 17 at 10:27

















up vote
1
down vote

favorite













This question is an exact duplicate of:




  • Finding an Inverse Function and Composition of Functions?

    1 answer




The functions of each pair are inverse to each other. For each pair, check that both compositions give the identity function.



$F: mathbb{R} to mathbb{R}$ and $F^{−1}:mathbb{R} to mathbb{R}$ are defined by $F(x)=3x+2$ and $F^{−1}(y)=dfrac{y−2}{3}$, for all $y in Bbb{R}.$



My attempt:



Inverse Function



For each particular but arbitrarily chosen $y in mathbb{R}$, according to the definition of $f^{-1}$, $f^{-1}(y) = dfrac{y-2}{3}$ is a unique real number $x$ such that $f(x) = y$.
begin{align*}
F(x) & = y\
3x + 2 & = y\
x & = frac{y-2}{3}
end{align*}

Therefore, $f^{-1}(y) = frac{y-2}{3}$.



Compositions of Functions.



The functions $g circ f$ and $f circ g$ are defined as follows:
$$(g circ f)(x) = g(f(x)) = g(3x + 2) = 3x + 2$$
for all $x in mathbb{Z}$.










share|cite|improve this question















marked as duplicate by N. F. Taussig, Community Nov 17 at 16:53


This question was marked as an exact duplicate of an existing question.















  • What is $g$ here?
    – drhab
    Nov 17 at 10:06










  • @drhab (g ◦ f )(x) = g( f (x)) for all x ∈ X, where g ◦ f is read “g circle f ” and g( f (x)) is read “g of f of x.” The function g ◦ f is called the composition of f and g.
    – adam sala
    Nov 17 at 10:10










  • I know, but in the last lines of your question $g$ just seems to fall from the sky. Further it is said that $g(3x+2)=3x+2$ which implies that $g$ is the identity. Is that really what you meant to say?
    – drhab
    Nov 17 at 10:13










  • yes @drhab , is the composition of function above correct?
    – adam sala
    Nov 17 at 10:23










  • If $f$ is prescribed by $xmapsto3x+2$ and $g$ is the identity function then $gcirc f$ is indeed prescribed by $xmapsto 3x+2$. Actually then $gcirc f=f$. But I do not see any functionality of this in this context.
    – drhab
    Nov 17 at 10:27















up vote
1
down vote

favorite









up vote
1
down vote

favorite












This question is an exact duplicate of:




  • Finding an Inverse Function and Composition of Functions?

    1 answer




The functions of each pair are inverse to each other. For each pair, check that both compositions give the identity function.



$F: mathbb{R} to mathbb{R}$ and $F^{−1}:mathbb{R} to mathbb{R}$ are defined by $F(x)=3x+2$ and $F^{−1}(y)=dfrac{y−2}{3}$, for all $y in Bbb{R}.$



My attempt:



Inverse Function



For each particular but arbitrarily chosen $y in mathbb{R}$, according to the definition of $f^{-1}$, $f^{-1}(y) = dfrac{y-2}{3}$ is a unique real number $x$ such that $f(x) = y$.
begin{align*}
F(x) & = y\
3x + 2 & = y\
x & = frac{y-2}{3}
end{align*}

Therefore, $f^{-1}(y) = frac{y-2}{3}$.



Compositions of Functions.



The functions $g circ f$ and $f circ g$ are defined as follows:
$$(g circ f)(x) = g(f(x)) = g(3x + 2) = 3x + 2$$
for all $x in mathbb{Z}$.










share|cite|improve this question
















This question is an exact duplicate of:




  • Finding an Inverse Function and Composition of Functions?

    1 answer




The functions of each pair are inverse to each other. For each pair, check that both compositions give the identity function.



$F: mathbb{R} to mathbb{R}$ and $F^{−1}:mathbb{R} to mathbb{R}$ are defined by $F(x)=3x+2$ and $F^{−1}(y)=dfrac{y−2}{3}$, for all $y in Bbb{R}.$



My attempt:



Inverse Function



For each particular but arbitrarily chosen $y in mathbb{R}$, according to the definition of $f^{-1}$, $f^{-1}(y) = dfrac{y-2}{3}$ is a unique real number $x$ such that $f(x) = y$.
begin{align*}
F(x) & = y\
3x + 2 & = y\
x & = frac{y-2}{3}
end{align*}

Therefore, $f^{-1}(y) = frac{y-2}{3}$.



Compositions of Functions.



The functions $g circ f$ and $f circ g$ are defined as follows:
$$(g circ f)(x) = g(f(x)) = g(3x + 2) = 3x + 2$$
for all $x in mathbb{Z}$.





This question is an exact duplicate of:




  • Finding an Inverse Function and Composition of Functions?

    1 answer








algebra-precalculus functions discrete-mathematics






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edited Nov 17 at 11:04









amWhy

191k27223439




191k27223439










asked Nov 17 at 9:49









adam sala

35




35




marked as duplicate by N. F. Taussig, Community Nov 17 at 16:53


This question was marked as an exact duplicate of an existing question.






marked as duplicate by N. F. Taussig, Community Nov 17 at 16:53


This question was marked as an exact duplicate of an existing question.














  • What is $g$ here?
    – drhab
    Nov 17 at 10:06










  • @drhab (g ◦ f )(x) = g( f (x)) for all x ∈ X, where g ◦ f is read “g circle f ” and g( f (x)) is read “g of f of x.” The function g ◦ f is called the composition of f and g.
    – adam sala
    Nov 17 at 10:10










  • I know, but in the last lines of your question $g$ just seems to fall from the sky. Further it is said that $g(3x+2)=3x+2$ which implies that $g$ is the identity. Is that really what you meant to say?
    – drhab
    Nov 17 at 10:13










  • yes @drhab , is the composition of function above correct?
    – adam sala
    Nov 17 at 10:23










  • If $f$ is prescribed by $xmapsto3x+2$ and $g$ is the identity function then $gcirc f$ is indeed prescribed by $xmapsto 3x+2$. Actually then $gcirc f=f$. But I do not see any functionality of this in this context.
    – drhab
    Nov 17 at 10:27




















  • What is $g$ here?
    – drhab
    Nov 17 at 10:06










  • @drhab (g ◦ f )(x) = g( f (x)) for all x ∈ X, where g ◦ f is read “g circle f ” and g( f (x)) is read “g of f of x.” The function g ◦ f is called the composition of f and g.
    – adam sala
    Nov 17 at 10:10










  • I know, but in the last lines of your question $g$ just seems to fall from the sky. Further it is said that $g(3x+2)=3x+2$ which implies that $g$ is the identity. Is that really what you meant to say?
    – drhab
    Nov 17 at 10:13










  • yes @drhab , is the composition of function above correct?
    – adam sala
    Nov 17 at 10:23










  • If $f$ is prescribed by $xmapsto3x+2$ and $g$ is the identity function then $gcirc f$ is indeed prescribed by $xmapsto 3x+2$. Actually then $gcirc f=f$. But I do not see any functionality of this in this context.
    – drhab
    Nov 17 at 10:27


















What is $g$ here?
– drhab
Nov 17 at 10:06




What is $g$ here?
– drhab
Nov 17 at 10:06












@drhab (g ◦ f )(x) = g( f (x)) for all x ∈ X, where g ◦ f is read “g circle f ” and g( f (x)) is read “g of f of x.” The function g ◦ f is called the composition of f and g.
– adam sala
Nov 17 at 10:10




@drhab (g ◦ f )(x) = g( f (x)) for all x ∈ X, where g ◦ f is read “g circle f ” and g( f (x)) is read “g of f of x.” The function g ◦ f is called the composition of f and g.
– adam sala
Nov 17 at 10:10












I know, but in the last lines of your question $g$ just seems to fall from the sky. Further it is said that $g(3x+2)=3x+2$ which implies that $g$ is the identity. Is that really what you meant to say?
– drhab
Nov 17 at 10:13




I know, but in the last lines of your question $g$ just seems to fall from the sky. Further it is said that $g(3x+2)=3x+2$ which implies that $g$ is the identity. Is that really what you meant to say?
– drhab
Nov 17 at 10:13












yes @drhab , is the composition of function above correct?
– adam sala
Nov 17 at 10:23




yes @drhab , is the composition of function above correct?
– adam sala
Nov 17 at 10:23












If $f$ is prescribed by $xmapsto3x+2$ and $g$ is the identity function then $gcirc f$ is indeed prescribed by $xmapsto 3x+2$. Actually then $gcirc f=f$. But I do not see any functionality of this in this context.
– drhab
Nov 17 at 10:27






If $f$ is prescribed by $xmapsto3x+2$ and $g$ is the identity function then $gcirc f$ is indeed prescribed by $xmapsto 3x+2$. Actually then $gcirc f=f$. But I do not see any functionality of this in this context.
– drhab
Nov 17 at 10:27












1 Answer
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If I understand well then you must check whether the functions $mathbb Rtomathbb R$ prescribed by $xmapsto3x+2$ and $xmapstofrac13(x-2)$ are inverses of each other.



That is indeed the case since their compositions are prescribed by: $xmapsto3x+2mapstofrac13((3x+2)-2)=x$ and: $xmapstofrac13(x-2)mapsto3(frac13(x-2))+2=x$ so both are identity functions.



This observation is enough to conclude that the functions are inverses of each other, and is what you are asked to do.






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    1 Answer
    1






    active

    oldest

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    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    0
    down vote



    accepted










    If I understand well then you must check whether the functions $mathbb Rtomathbb R$ prescribed by $xmapsto3x+2$ and $xmapstofrac13(x-2)$ are inverses of each other.



    That is indeed the case since their compositions are prescribed by: $xmapsto3x+2mapstofrac13((3x+2)-2)=x$ and: $xmapstofrac13(x-2)mapsto3(frac13(x-2))+2=x$ so both are identity functions.



    This observation is enough to conclude that the functions are inverses of each other, and is what you are asked to do.






    share|cite|improve this answer

























      up vote
      0
      down vote



      accepted










      If I understand well then you must check whether the functions $mathbb Rtomathbb R$ prescribed by $xmapsto3x+2$ and $xmapstofrac13(x-2)$ are inverses of each other.



      That is indeed the case since their compositions are prescribed by: $xmapsto3x+2mapstofrac13((3x+2)-2)=x$ and: $xmapstofrac13(x-2)mapsto3(frac13(x-2))+2=x$ so both are identity functions.



      This observation is enough to conclude that the functions are inverses of each other, and is what you are asked to do.






      share|cite|improve this answer























        up vote
        0
        down vote



        accepted







        up vote
        0
        down vote



        accepted






        If I understand well then you must check whether the functions $mathbb Rtomathbb R$ prescribed by $xmapsto3x+2$ and $xmapstofrac13(x-2)$ are inverses of each other.



        That is indeed the case since their compositions are prescribed by: $xmapsto3x+2mapstofrac13((3x+2)-2)=x$ and: $xmapstofrac13(x-2)mapsto3(frac13(x-2))+2=x$ so both are identity functions.



        This observation is enough to conclude that the functions are inverses of each other, and is what you are asked to do.






        share|cite|improve this answer












        If I understand well then you must check whether the functions $mathbb Rtomathbb R$ prescribed by $xmapsto3x+2$ and $xmapstofrac13(x-2)$ are inverses of each other.



        That is indeed the case since their compositions are prescribed by: $xmapsto3x+2mapstofrac13((3x+2)-2)=x$ and: $xmapstofrac13(x-2)mapsto3(frac13(x-2))+2=x$ so both are identity functions.



        This observation is enough to conclude that the functions are inverses of each other, and is what you are asked to do.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 17 at 10:03









        drhab

        95.1k543126




        95.1k543126















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