# Thread: Show that f o g is one-to-one

1. ## Show that f o g is one-to-one

(a) Show that the composition of two one-to-one functions, f and g, is one-to-one

(b) Express $\displaystyle (f o g)^{-1}$in terms of $\displaystyle f^{-1} and g^{-1}$.

2. Dear 450081592,

1) Suppose a,b $\displaystyle \in{Dom (f_og)}$ such that,

$\displaystyle f_o{g(a)}=p$ and $\displaystyle f_o{g(b)}=p$

Then, f[g(a)]=f[g(b)]

g(a)=g(b) since f is an one to one function.

a=b since g is an one to one function.

Therefore $\displaystyle f_o{g(a)}=f_o{g(b)}\Rightarrow{a=b}$

Hence $\displaystyle f_o{g}$ is an one to one function.

2) Supose g(a) = c and f(c) = b,

Therefore, $\displaystyle f_og(a)=b$

Since $\displaystyle f_og$ is one to one it is invertible,

$\displaystyle (f_og)^{-1}(b)=a$-----------A

Since f and g are one to one they are invertible.

$\displaystyle g^{-1}(c)=a$ and $\displaystyle f^{-1}(b)=c$

Therefore $\displaystyle g^{-1}[f^{-1}(b)]=a$-------B

From A and B,

$\displaystyle g^{-1}_of^{-1}(b)=(f_og)^{-1}(b)$ $\displaystyle \forall$ $\displaystyle b\in Dom{(f_og)}$

3. Originally Posted by 450081592
(a) Show that the composition of two one-to-one functions, f and g, is one-to-one

(b) Express $\displaystyle (f o g)^{-1}$in terms of $\displaystyle f^{-1} and g^{-1}$.
Originally Posted by Sudharaka
Dear 450081592,

1) Suppose a,b $\displaystyle \in{Dom (f_og)}$ such that,

$\displaystyle f_o{g(a)}=p$ and $\displaystyle f_o{g(b)}=p$

Then, f[g(a)]=f[g(b)]

g(a)=g(b) since f is an one to one function.

a=b since g is an one to one function.

Therefore $\displaystyle f_o{g(a)}=f_o{g(b)}\Rightarrow{a=b}$

Hence $\displaystyle f_o{g}$ is an one to one function.

2) Supose g(a) = c and f(c) = b,

Therefore, $\displaystyle f_og(a)=b$

Since $\displaystyle f_og$ is one to one it is invertible,

$\displaystyle (f_og)^{-1}(b)=a$-----------A

Since f and g are one to one they are invertible.

$\displaystyle g^{-1}(c)=a$ and $\displaystyle f^{-1}(b)=c$

Therefore $\displaystyle g^{-1}[f^{-1}(b)]=a$-------B

From A and B,

$\displaystyle g^{-1}_of^{-1}(b)=(f_og)^{-1}(b)$ $\displaystyle \forall$ $\displaystyle b\in Dom{(f_og)}$
Jeez you two, haha. Try... \circ $\displaystyle f\circ g$

Also, for the second one there is a particularly nice result if your functions are both mappings from a set $\displaystyle X$ into itself. It follows that $\displaystyle f,g\in S_X$ (the permutation group on $\displaystyle X$) from where it follows from basic group theory that $\displaystyle \left(ab\right)^{-1}=b^{-1}a^{-1}$.

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### prove that the function fog is one-to-one

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