To find [tex]\(x\)[/tex] given that [tex]\(f(x) = 7x^3 + 2x + 2\)[/tex] and [tex]\(f^{-1}(x) = 1\)[/tex], we start by understanding the meaning of the inverse function condition. Specifically, [tex]\(f^{-1}(1) = x\)[/tex] means that [tex]\(f(x) = 1\)[/tex].
So, we need to solve the equation:
[tex]\[ f(x) = 1 \][/tex]
Given the function [tex]\( f(x) = 7x^3 + 2x + 2\)[/tex], we set [tex]\( f(x) \)[/tex] equal to 1:
[tex]\[ 7x^3 + 2x + 2 = 1 \][/tex]
This simplifies to:
[tex]\[ 7x^3 + 2x + 1 = 0 \][/tex]
Now, we need to find the roots of the cubic equation:
[tex]\[ 7x^3 + 2x + 1 = 0 \][/tex]
The solutions to this cubic equation are:
[tex]\[ x = \frac{2}{7\left(-\frac{1}{2} - \frac{\sqrt{3}i}{2}\right) \left(\frac{27}{14} + \frac{3\sqrt{4641}}{98}\right)^{1/3}} - \frac{\left(-\frac{1}{2} - \frac{\sqrt{3}i}{2}\right) \left(\frac{27}{14} + \frac{3\sqrt{4641}}{98}\right)^{1/3}}{3} \][/tex]
[tex]\[ x = -\frac{\left(-\frac{1}{2} + \frac{\sqrt{3}i}{2}\right) \left(\frac{27}{14} + \frac{3\sqrt{4641}}{98}\right)^{1/3}}{3} + \frac{2}{7\left(-\frac{1}{2} + \frac{\sqrt{3}i}{2}\right) \left(\frac{27}{14} + \frac{3\sqrt{4641}}{98}\right)^{1/3}} \][/tex]
[tex]\[ x = -\frac{\left(\frac{27}{14} + \frac{3\sqrt{4641}}{98}\right)^{1/3}}{3} + \frac{2}{7 \left(\frac{27}{14} + \frac{3\sqrt{4641}}{98}\right)^{1/3}} \][/tex]
Therefore, the values of [tex]\(x\)[/tex] that satisfy [tex]\(f^{-1}(1)\)[/tex] are:
[tex]\[ x = \frac{2}{7\left(-\frac{1}{2} - \frac{\sqrt{3}i}{2}\right) \left(\frac{27}{14} + \frac{3\sqrt{4641}}{98}\right)^{1/3}} - \frac{\left(-\frac{1}{2} - \frac{\sqrt{3}i}{2}\right) \left(\frac{27}{14} + \frac{3\sqrt{4641}}{98}\right)^{1/3}}{3} \][/tex]
[tex]\[ x = -\frac{\left(-\frac{1}{2} + \frac{\sqrt{3}i}{2}\right) \left(\frac{27}{14} + \frac{3\sqrt{4641}}{98}\right)^{1/3}}{3} + \frac{2}{7\left(-\frac{1}{2} + \frac{\sqrt{3}i}{2}\right) \left(\frac{27}{14} + \frac{3\sqrt{4641}}{98}\right)^{1/3}} \][/tex]
[tex]\[ x = -\frac{\left(\frac{27}{14} + \frac{3\sqrt{4641}}{98}\right)^{1/3}}{3} + \frac{2}{7 \left(\frac{27}{14} + \frac{3\sqrt{4641}}{98}\right)^{1/3}} \][/tex]
These are the values of [tex]\(x\)[/tex] where [tex]\(f(x) = 1\)[/tex], which corresponds to [tex]\(f^{-1}(1)\)[/tex].
