Question

Solve y=f(x) for x. Then find the input when the output is -3.

f(x) = (x-5)^3 -1
x = __
The input is __ when the output is -3.

Answer 1

Please check the explanation

Explanation:

Given the function

`f\left(x\right)\:=\:\left(x-5\right)^3-1`

Given that the output = -3

i.e. y = -3

now substituting the value y=-3 and solve for x to determine the input 'x'

`\:\:y=\:\left(x-5\right)^3-1`

`-3\:=\:\left(x-5\right)^3-1\:\:\:`

switch sides

`\left(x-5\right)^3-1=-3`

Add 1 to both sides

`\left(x-5\right)^3-1+1=-3+1`

`\left(x-5\right)^3=-2`

`\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)}(-1-√(3)i)/(2),\:\sqrt[3]{f\left(a\right)}(-1+√(3)i)/(2)`

Thus, the input values are:

`x=-\sqrt[3]{2}+5,\:x=\frac{\sqrt[3]{2}\left(1+5\cdot \:2^{(2)/(3)}\right)}{2}-i\frac{\sqrt[3]{2}√(3)}{2},\:x=\frac{\sqrt[3]{2}\left(1+5\cdot \:2^{(2)/(3)}\right)}{2}+i\frac{\sqrt[3]{2}√(3)}{2}`

And the real input is:

`x=-\sqrt[3]{2}+5`

• 
  `x=3.74`