Question

homework assignment 1.7: inverse functions
score: 8.5/15 answered: 9/15
question 10
find the inverse function of $f(x) = 8 + \sqrt3{x}$.
$f^{-1}(x) = $
question help: video written example

Explanation:

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

We start with the function \( f(x) = 8+\sqrt[3]{x} \). Replace \( f(x) \) with \( y \), so we have \( y = 8+\sqrt[3]{x} \).

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

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x = 8+\sqrt[3]{y} \).

Step3: Solve for \( y \)

First, subtract 8 from both sides of the equation: \( x - 8=\sqrt[3]{y} \). Then, to eliminate the cube root, we cube both sides. Cubing the left side gives \( (x - 8)^3 \), and cubing the right side gives \( y \) (since \( (\sqrt[3]{y})^3=y \)). So we have \( y=(x - 8)^3 \).

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

Now, replace \( y \) with \( f^{-1}(x) \) to get the inverse function. So \( f^{-1}(x)=(x - 8)^3 \).

Answer:

\( (x - 8)^3 \)