consider the function (f o g)^(-1) (x)  found by first finding the composite of f and g and then taking the inverse of that composite. Will (f o g)^(-1) (x) = f^(-1) (x) o g ^(-1) or will (f o g)^(-1) (x) = g^(-1) (x) o f^(-1) (x) or is neither statment true? Justify your answer using the following two set of functions: f(x) = x-2 g(x)=x^2   f(x) = 1/x+1 g(x)=3x-2

Expert Answers

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Note: `f@g(x) = f(g(x))`

This means, where you used to see an "x" in the equation for f(x), now plug in "g(x)"

To find `f^(-1)(x)`

First start with your equation, then switch the x and the y, then solve for y.

So for example:


Think of this as: `y=4x+3`

To find the inverse, switch the x and the y:


Then solve for y


With this in mind, start with the first pair of functions:







`f^(-1) @ g^(-1) (x) = root()(x)+2`


So, at least in this case, 



Do the same for the second set of equations:





` `

` `





` `

So again, we have:



PS: We have only shown 

` (f@g)^(-1)(x)=g^(-1)@f^(-1)(x) ` 

is true for these pairs of functions, but it is always true

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