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Thread: Inverse function

  1. #1
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    Inverse function

    The problem says; find and verify the inverse function of f(x)=2x^3 - 4

    I'm stuck on what to do first Any help would be appreciated!
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    Let $\displaystyle y = f^{-1}(x)$. Then switch the positions of x and f(x) and change f(x) to y (since doing this switch changes f(x)).

    $\displaystyle x = 2y^{3} - 4$

    Solve for y (i.e. $\displaystyle f^{-1}(x)$) in terms of x
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    MHF Contributor arbolis's Avatar
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    find and verify the inverse function of f(x)=2x^3 - 4
    Okay. You know that $\displaystyle f(f^{-1}(x))=y$.
    The idea to find the inverse function of f, that is $\displaystyle f^{-1}$, is to get x in terms of y. Let $\displaystyle y=f(x)$, then we have $\displaystyle y=2x^3-4$. Add 4 in each sides of the equation, we get $\displaystyle y+4=2x^3$. Divide by 2 now, you get $\displaystyle \frac{y+4}{2}=x^3$. Now take the 3rd root in each sides. You get finally that $\displaystyle (\frac{y+4}{2})^\frac{1}{3}=x$, or $\displaystyle x=(\frac{y}{2}+2)^\frac{1}{3}$.
    To verify it, just replace the x you had in your first equation by $\displaystyle (\frac{y}{2}+2)^\frac{1}{3}$ and if you find that it's equal to y, that means you did it.
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