Respuesta :

dashataran

Answer:

[tex]g^{-1}(x)=(x+2)^3+2[/tex]

Step-by-step explanation:

[tex]\text{Solution: }\\\\g(x)=\sqrt[3]{x-2}-2\\\\\text{or, }y=\sqrt[3]{x-2}-2\\\\\text{or, }y+2=\sqrt[3]{x-2}\\\\\text{Cubing both sides,}\\\\\text{}\hspace{0.5cm}(y+2)^3=x-2\\\\\text{Interchanging }x\text { and }y,\\\\(x+2)^3=y-2\\\\\text{or, }y=(x+2)^3+2\\\\\therefore\ g^{-1}(x)=(x+2)^3+2[/tex]

Heather

Answer: g'(x) = (x + 2)³ + 2

Step-by-step explanation:

      To find the inverse of the given function, we will set the function equal to y, flip the x and y variables (x, y to y, x), and solve for y again.

Given:

  [tex]\displaystyle g(x)= \sqrt[3]{x-2} -2[/tex]

Set equal to y:

  [tex]\displaystyle y= \sqrt[3]{x-2} -2[/tex]

Swap x and y variables:

  [tex]\displaystyle x= \sqrt[3]{y-2} -2[/tex]

Add 2 to both sides of the equation:

  [tex]\displaystyle x+2= \sqrt[3]{y-2}[/tex]

Cube both sides of the equation:

  [tex]\displaystyle (x+2)^3=y-2[/tex]

Add 2 to both sides of the equation:

  [tex]\displaystyle (x+2)^3+2=y[/tex]

Inverse notation:

  [tex]\displaystyle g'(x)=(x+2)^3+2[/tex]

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