Inverse

An function’s inverse is founded by the swapping the input and output values.

Therefore, if the original function have a output value of y, by pug in y into the inverse function. It will yields out the input of the original function that yield out y.

Requirement to become an Inverse

One to One

That the y value only match one x value, and x value only match one y value

Continuous

No x value will yield out undefined point.

Notation

$$ \text {g(x) is inverse of f(x)}↔g(x)=f^{-1}(x)↔ f(g(x))=x=f(g(x)) $$

<aside> 💡 Inverse function at corresponding points, have reciprocal slope

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Derivative of the Inverse

$$ f'^{-1}(x)=\frac 1 {f'(f^{-1}(x))} $$

How to derivative it

1. We have the equation $f(f^{-1}(x))=x$ That they cancels each other out
2. We will take the derivative (implicit differentiation) of them $f'(f^{-1}(x))f'^{-1}(x)=1$
3. isolate the derivative of the inverse. $f'^{-1}(x)=\frac 1 {f'(f^{-1}(x))}$

Or you can use the implicit differentiation after you swapped the x and y value.

<aside> 💡 Remember: if a question ask you for the equation for the tangent line of the inverse of the function, the x and y point (original) didn’t change.

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