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Numerade Educator

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Problem 59 Hard Difficulty

Graph the function $ f(x) = \sqrt{x^3 + x^2 + x +1} $ and explain why it is one-to-one. Then use a computer algebra system to find an explicit expression for $ f^{-1} (x) $. (Your CAS will produce three possible expressions. Explain why two of them are irrelevant in this context.)

Answer

$$\frac{1}{6} \frac{M^{2 / 3}-8-2 M^{1 / 3}}{2 M^{1 / 3}}$$
$$M=108 x^{2}+12 \sqrt{48-120 x^{2}+81 x^{4}}-80$$.

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Video Transcript

it's clear. So when you read here, So when we grab up of X is equal to the square root of ex cute less X Square plus X plus one, we get a 1 to 1 function that looks like this. It's 1 to 1, because for every X value there is at most one y value and it also passes the horizontal line does. And we know that when we use the computer algebra system to solve for why we get three solutions, but we can't use two of them since they contain I, which are which make them non real solutions. So are inverse. Function becomes you. Brute of 27 X square last three square root of three. Where were you of 27 x to the fore minus 40 X square plus 16 minus 20 all over three times Cube root of to minus two times cube root of to all over three Q. Brute of 27 X Square plus three square root of three. We're a Route 27 next before minus 40 X Square plus 16 minus 20 minus 1/3