Answer:
Please check the explanation
Step-by-step explanation:
Given the function
[tex]f\left(x\right)\:=\:\left(x-5\right)^3-1[/tex]
Given that the output = -3
i.e. y = -3
now substituting the value y=-3 and solve for x to determine the input 'x'
[tex]\:\:y=\:\left(x-5\right)^3-1[/tex]
[tex]-3\:=\:\left(x-5\right)^3-1\:\:\:[/tex]
switch sides
[tex]\left(x-5\right)^3-1=-3[/tex]
Add 1 to both sides
[tex]\left(x-5\right)^3-1+1=-3+1[/tex]
[tex]\left(x-5\right)^3=-2[/tex]
[tex]\mathrm{For\:}g^3\left(x\right)=f\left(a\right)\mathrm{\:the\:solutions\:are\:}g\left(x\right)=\sqrt[3]{f\left(a\right)},\:\sqrt[3]{f\left(a\right)}\frac{-1-\sqrt{3}i}{2},\:\sqrt[3]{f\left(a\right)}\frac{-1+\sqrt{3}i}{2}[/tex]
Thus, the input values are:
[tex]x=-\sqrt[3]{2}+5,\:x=\frac{\sqrt[3]{2}\left(1+5\cdot \:2^{\frac{2}{3}}\right)}{2}-i\frac{\sqrt[3]{2}\sqrt{3}}{2},\:x=\frac{\sqrt[3]{2}\left(1+5\cdot \:2^{\frac{2}{3}}\right)}{2}+i\frac{\sqrt[3]{2}\sqrt{3}}{2}[/tex]
And the real input is:
[tex]x=-\sqrt[3]{2}+5[/tex]
