Questions: Find the inverse of f(x)=5x^3+11. a.) f^(-1)(x)=sqrt((x+11)/5) b.) f^(-1)(x)=sqrt((x-11)/5) c.) f^(-1)(x)=sqrt[3]((x+11)/5) d.) f^(-1)(x)=sqrt[3]((x-11)/5)

Transcript text: Find the inverse of $f(x)=5 x^{3}+11$. a.) $f^{-1}(x)=\sqrt{\frac{x+11}{5}}$ b.) $f^{-1}(x)=\sqrt{\frac{x-11}{5}}$ c.) $f^{-1}(x)=\sqrt[3]{\frac{x+11}{5}}$ d.) $f^{-1}(x)=\sqrt[3]{\frac{x-11}{5}}$

Solution

Solution Steps

Step 1: Replace $ f(x) $ with $ y $

Start by writing the function as: \[ y = 5x^{3} + 11 \]

Step 2: Swap $ x $ and $ y $

Swap $ x $ and $ y $ to find the inverse: \[ x = 5y^{3} + 11 \]

Step 3: Solve for $ y $

Subtract 11 from both sides: \[ x - 11 = 5y^{3} \] Divide both sides by 5: \[ \frac{x - 11}{5} = y^{3} \] Take the cube root of both sides: \[ y = \sqrt[3]{\frac{x - 11}{5}} \]

Step 4: Write the inverse function

The inverse function is: \[ f^{-1}(x) = \sqrt[3]{\frac{x - 11}{5}} \]