Find f superscript negative 1 baseline for the function lower f left parenthesis x right parenthesis equals root index 3 Start Root x minus 2 End Root plus 8
To find \(f^{-1}(x)\) for the function \(f(x) = \sqrt[3]{x-2} + 8\), we follow these steps:
Step 1: Replace \(f(x)\) with \(y\)
\[y = \sqrt[3]{x-2} + 8\]
Step 2: Swap \(x\) and \(y\)
\[x = \sqrt[3]{y-2} + 8\]
Step 3: Solve for \(y\)
Subtracting 8 from both sides, we get:
\[x - 8 = \sqrt[3]{y-2}\]
Cubing both sides:
\[(x - 8)^3 = y - 2\]
Adding 2 to both sides:
\[(x - 8)^3 + 2 = y\]
Step 4: Replace \(y\) with \(f^{-1}(x)\)
\[\begin{aligned} f^{-1}(x) &= (x - 8)^3 + 2 \end{aligned}\]
Therefore, \(f^{-1}(x) = (x - 8)^3 + 2\) for the given function \(f(x) = \sqrt[3]{x-2} + 8\).
To find the inverse function f^(-1) for the given function f(x) = √(3√(x-2)) + 8, follow these steps:
Step 1: Replace f(x) with y.
y = √(3√(x-2)) + 8
Step 2: Swap x and y.
x = √(3√(y-2)) + 8
Step 3: Isolate the radical expression.
x - 8 = √(3√(y-2))
Step 4: Square both sides to remove the outermost radical.
(x - 8)^2 = 3√(y-2)
Step 5: Cube both sides to remove the remaining radical.
[(x - 8)^2]^3 = (y-2)
Step 6: Simplify the expression.
(x - 8)^6 = y - 2
Step 7: Solve for y to find the inverse function.
y = (x - 8)^6 + 2
Therefore, the inverse function f^(-1)(x) is given by:
f^(-1)(x) = (x - 8)^6 + 2