Question

for the function $f(x) = sqrt3{x + 9}$, find $f^{-1}(x)$.
answer

• $f^{-1}(x) = sqrt3{x} - 9$
• $f^{-1}(x) = sqrt3{x - 9}$
• $f^{-1}(x) = (x + 9)^3$
• $f^{-1}(x) = x^3 - 9$

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Explanation:

Step1: Replace \( f(x) \) with \( y \)

We start with the function \( f(x)=\sqrt[3]{x + 9} \). Replace \( f(x) \) with \( y \), so we have \( y=\sqrt[3]{x + 9} \).

Step2: Swap \( x \) and \( y \)

To find the inverse function, we swap the roles of \( x \) and \( y \). This gives us \( x=\sqrt[3]{y + 9} \).

Step3: Solve for \( y \)

To solve for \( y \), we first cube both sides of the equation to eliminate the cube root. Cubing both sides, we get \( x^{3}=(\sqrt[3]{y + 9})^{3} \). Since \( (\sqrt[3]{a})^{3}=a \) for any real number \( a \), this simplifies to \( x^{3}=y + 9 \). Then, we subtract 9 from both sides to solve for \( y \): \( y=x^{3}-9 \).

Step4: Replace \( y \) with \( f^{-1}(x) \)

We replace \( y \) with \( f^{-1}(x) \), so the inverse function is \( f^{-1}(x)=x^{3}-9 \).

Answer:

\( f^{-1}(x) = x^{3}-9 \) (the fourth option: \( f^{-1}(x) = x^{3}-9 \))