For the function $y = f (x) = 2 x^{3} - 9$ :
b. Find a formula for $x = f^{- 1} (y)$ .

Answer

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Answer

$f^{- 1} (y) = {(\frac{y + 9}{2})}^{\frac{1}{3}}$ is the final answer.

Steps

Step 1 ：Given the function $y = f (x) = 2 x^{3} - 9$

Step 2 ：To find the inverse of a function, we need to switch the roles of x and y and solve for y. In this case, we need to solve the equation $y = 2 x^{3} - 9$ for x.

Step 3 ：The inverse function of $f (x)$ is a cubic root function. However, the cubic root function has three solutions in the complex plane, two of which are complex. Since the original function $f (x)$ is real-valued, we only consider the real solution for the inverse function.

Step 4 ： $f^{- 1} (y) = {(\frac{y + 9}{2})}^{\frac{1}{3}}$ is the final answer.

For the function y=f(x)=2x3−9 :b. Find a formula for x=f−1(y).

Answer

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For the function $y = f (x) = 2 x^{3} - 9$ :
b. Find a formula for $x = f^{- 1} (y)$ .