If $f$ is one-to-one, find an equation for its inverse.
\[
f(x)=4 x^{3}-3
\]
A. $f^{-1}(x)=\sqrt[3]{\frac{x}{4}}+3$
B. $f^{-1}(x)=\sqrt[3]{\frac{x+3}{4}}$
C. $f^{-1}(x)=\sqrt[3]{\frac{x-3}{4}}$
D. The function is not one-to-one.
Answer
So, the inverse function is \(f^{-1}(x)=\sqrt[3]{\frac{x+3}{4}}\), which corresponds to option B.
Step 1 ：To find the inverse of the function \(f(x)=4x^3-3\), we first replace \(f(x)\) with \(y\), so the equation becomes \(y=4x^3-3\).
Step 2 ：Next, we swap \(x\) and \(y\) to get \(x=4y^3-3\).
Step 3 ：Then, we solve for \(y\) to get the inverse function. Add 3 to both sides to get \(x+3=4y^3\).
Step 4 ：Divide both sides by 4 to get \(\frac{x+3}{4}=y^3\).
Step 5 ：Finally, take the cube root of both sides to get \(y=\sqrt[3]{\frac{x+3}{4}}\).
Step 6 ：So, the inverse function is \(f^{-1}(x)=\sqrt[3]{\frac{x+3}{4}}\), which corresponds to option B.
