f(x)=10x-10 find the value of (f^-1*f)(10)

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lfryerda eNotes educator| Certified Educator

The composition of an inverse and its function is just x.  That is `f^{-1} circ f(x)=x` .  In this case, this means that for `f(x)=10x-10` then `f^{-1} circ f(10)=10` .    We can verify this by calculating the inverse through interchanging x and y then solving for y:

`y=10x-10`   interchange x and y

`x=10y-10`   isolate y

`x+10=10y`   divide by 10


This means that the inverse function is `f^{-1}(x)={x+10}/10` .

Now calculate explicitly the composition:

`f^{-1} circ f(10)`


`=f^{-1}(10(10)-10)`   simplify the argument

`=f^{-1}(100-10)`  further simplify

`=f^{-1}(90)`   now sub into the inverse

`={90+10}/10`  simplify


The composition of the function and its inverse is 10.

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