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Numerade Educator

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Problem 9 Easy Difficulty

Finding an Inverse Function Informally In
Exercises $7-12$ , find the inverse function of $f$ informally.
Verify that $f\left(f^{-1}(x)\right)=x$ and $f^{-1}(f(x))=x$
$$f(x)=3 x+1$$

Answer

$f(x)=3 x+1$
$f^{-1}(x)=\frac{x-1}{3}$
$f\left(f^{-1}(x)\right)=f\left(\frac{x-1}{3}\right)=3\left(\frac{x-1}{3}\right)+1=x$
$f^{-1}(f(x))=f^{-1}(3 x+1)=\frac{(3 x+1)-1}{3}=x$

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Video Transcript

Yeah, were given the function. F of X is equal to three x plus one. We wanna informally find the inverse function, but we have to think about undoing this function so toe undo taking an input, multiplying by three and adding one. We would first subtract one because that's the opposite. And then divide by three. We're going a complete opposite direction and that the way that would look when we want to subtract one and then divide by three would be taken in input, subtracting one and then dividing the whole thing. 53 Let's verify this by composing F of f inverse of X would be f of X minus 1/3, which would be three times X minus 1/3 plus one with the threes will cancel. And so that's X minus one plus one or X. The other direction F inverse of f of X would be f inverse of three X plus one, which would be three x plus one minus 1/3 or three X over three, which is X. Yes, so we just verified that it is indeed the inverse. Okay,