Question

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

Short answer

The explicit function is \({f^{ - 1}}\left( x \right) = - \frac{{\sqrt[3]{4}}}{6}\left( {\sqrt(3){{D - 27{x^2} + 20}} - \sqrt[3]{{D + 27{x^2} - 20}} + \sqrt[3]{2}} \right)\) with \(D = 3\sqrt 3 \sqrt {27{x^4} - 40{x^2} + 16} \).

Step-by-step solution

Graph the function \(f\left( x \right) = \sqrt {{x^3} + {x^2} + x + 1} \) and explain why it is one-to-one

Sketch the graph of the function \(f\left( x \right) = \sqrt {{x^3} + {x^2} + x + 1} \) as shown below:

It is observed from the graph of \(y = f\left( x \right) = \sqrt {{x^3} + {x^2} + x + 1} \) is the increasing function, therefore, \(f\) is a one-to-one function.

Use a computer algebra system to find an explicit function

The procedure to use a computer algebra system to find an explicit function is as follows:

1. Open the computer algebra system calculator. Enter the equation\(\sqrt {{x^3} + {x^2} + x + 1} \)in the tab.
2. Enter the “CALCULATE” button in the CAS calculator.

There will probably be two (irrelevant) solutions that involve imaginary expressions as well as one that can be simplified as shown below:

\(\begin{aligned}y &= {f^{ - 1}}\left( x \right)\\ &= - \frac{{\sqrt(3){4}}}{6}\left( {\sqrt(3){{D - 27{x^2} + 20}} - \sqrt(3){{D + 27{x^2} - 20}} + \sqrt(3){2}} \right)\end{aligned}\)

Here, \(D = 3\sqrt 3 \sqrt {27{x^4} - 40{x^2} + 16} \) or it is equivalent to \(\frac{1}{6}\frac{{{M^{\frac{2}{3}}} - 8 - 2{M^{\frac{1}{3}}}}}{{2{M^{\frac{1}{3}}}}}\) where \(M = 108{x^2} + 12\sqrt {48 - 120{x^2} + 81{x^4}} - 80\).