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Find the Inverse of a Cubic Function

A step by step interactive worksheet to be used to develop the skill of fincding the inverse of cubic functions is presented. As many questions, including solutions, may be generated interactively.

QUESTION: Find the inverse of the function $f(x)=-\dfrac{2}{7}\left(x+2\right)^3+1$.
Step by step solution

STEP 1: Write the function as an equation replacing

$f(x)$ by

$y$ .

$y=-\dfrac{2}{7}\left(x+2\right)^3+1$

STEP 2: Solve for

$x$ the equation obtained in step 1.

$y-1=-\dfrac{2}{7}\left(x+2\right)^3$

$-\dfrac{7}{2}y+\dfrac{7}{2}=\left(x+2\right)^3$

$x+2=\sqrt[3]{-\dfrac{7}{2}y+\dfrac{7}{2}}$

$x=\left(\sqrt[3]{-\dfrac{7}{2}y+\dfrac{7}{2}}-2\right)$

STEP 3: Interchange

$x$ and

$y$ in the above equation.

$y=\left(\sqrt[3]{-\dfrac{7}{2}x+\dfrac{7}{2}}-2\right)$

STEP 4: Write the inverse function.

$f^{-1}(x) = y = \left( \sqrt[3]{-\dfrac{7}{2}x+\dfrac{7}{2}}-2\right)$