Question

For the function f(x)

Answer 1

Given:

The function is

$f(x)=\dfrac{8\sqrt[3]{x}-8}{9}$

To find:

The inverse function $f^{-1}(x)$.

Solution:

We have,

$f(x)=\dfrac{8\sqrt[3]{x}-8}{9}$

Step 1: Putting f(x)=y, we get

$y=\dfrac{8\sqrt[3]{x}-8}{9}$

Step 2: Interchange x and y.

$x=\dfrac{8\sqrt[3]{y}-8}{9}$

Step 3: Isolate variable y.

$9x=8\sqrt[3]{y}-8$

$9x+8=8\sqrt[3]{y}$

$\dfrac{9x+8}{8}=\sqrt[3]{y}$

Taking cube on both sides, we get

$\left(\dfrac{9x+8}{8}\right)^3=y$

$y=\left(\dfrac{9x+8}{8}\right)^3$

Step 4: Putting $y=f^{-1}(x)$, we get

$f^{-1}(x)=\left(\dfrac{9x+8}{8}\right)^3$

Therefore, the correct option is C.