Let $f(x) = 7 + 5x^3 + x^7$.
What will the inverse of $f(x)$ be?
How do I isolate $x$? I'm not being able to group $x$ together
And then how do I find $(f^{-1})''(1)$?
2026-04-02 10:02:32.1775124152
How can I find the inverse of $f(x) = 7 + 5x^3 + x^7$?
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You may not be able to find a simple formula for the inverse. You would have to be able to solve $y = 7 + 5x^3 + x^7$ for $x$ in terms of $y$. Not every algebraic equation can be solved in terms of elementary functions. However, you can find $f^{-1}(1)$ by finding an $x$ such that $7 + 5x^3 + x^7 = 1$. For that you can use the rational zeroes theorem to see if the equation has any rational zeros. In fact, $x=-1$ works. So, $f^{-1}(1)=-1$.
Another question of interest is whether that function even has an inverse. If you're in a calculus course you can take its derivative to show it is a strictly increasing function and hence is one-to-one and has an inverse.