Given the one-to-one function $f(x)=x^{3}+6$ find the following. Hint: You do not need to find the equation for $f^{-1}(x)$.
a. $f(-4)$
b. $f^{-1}(-58)$
a. Find the following value.
\[
f(-4)=
\]
b. Find the following value.
\[
f^{-1}(-58)=
\]
Answer
Final Answer: \(b. f^{-1}(-58)=\boxed{-4}\)
Step 1 ：Given the one-to-one function \(f(x)=x^{3}+6\), we need to find the following values.
Step 2 ：For part a, we need to substitute \(x=-4\) into the function \(f(x)=x^{3}+6\) and calculate the result.
Step 3 ：Substituting \(x=-4\) into the function, we get \(f(-4)=(-4)^{3}+6=-58\).
Step 4 ：For part b, we need to solve the equation \(f(x)=-58\) for \(x\). Since \(f(x)\) is a one-to-one function, there will be a unique solution.
Step 5 ：Solving the equation \(f(x)=-58\) for \(x\), we get \(x=-4\).
Step 6 ：Final Answer: \(a. f(-4)=\boxed{-58}\)
Step 7 ：Final Answer: \(b. f^{-1}(-58)=\boxed{-4}\)
