Find the equation of the inverse function for the following function `f(X)=3x+1` and `g(x)=x+2` and find `(f/g)(x)` and `(f/g)(1)`

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should substitute y for `f(x)`  and `g(x)`  such that:

`y = 3x + 1 => y - 1 = 3x => x = y/3 - 1/3`

Hence, the equation of inverse to the function `f(x)`  is `f^(-1)(x) = x/3 - 1/3` .

`y = x + 2 => x = y - 2`

Hence, the equation of inverse to the function `g(x)`  is `g^(-1)(x) =x-2` .

You need to evaluate `(f/g)(x) = (f(x))/(g(x))`  such that:

`(f/g)(x) = (3x + 1)/(x + 2)`

You need to evaluate `(f/g)(x)`  at `x = 1`  such that:

`(f/g)(1) = (3*1 + 1)/(1 + 2) => (f/g)(1) =4/3`

Hence, evaluating the inverses to the function `f(x)`  and `g(x)`  yields  `f^(-1)(x) = x/3 - 1/3`  and `g^(-1)(x) = x-2`  and evaluating `(f/g)(x)`  and `(f/g)(1)`  yields `(f/g)(x) = (3x + 1)/(x + 2) ` and `(f/g)(1) = 4/3` .

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tiburtius | High School Teacher | (Level 2) Educator

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By definition if `f^(-1)` is inverse of `f` then `f(f^(-1)(x))=x.`

Inverse of `f:`

`3f^(-1)(x)+1=x`

Next we proceed as if we are solving a linear equation.

`3f^(-1)(x)=x-1`

`f^(-1)(x)=(x-1)/(3)`

Similary inverse of `g:`

`g^(-1)(x)+2=x`

`g^(-1)(x)=x-2`

By definition `(f/g)(x)=(f(x))/(g(x))` hence

`(f/g)(x)=(3x+1)/(x+2)`.

So `(f/g)(1)=(3cdot1+1)/(1+2)=4/3`.

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