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

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

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