Questions: Given f(x)=(x-2)^3+5, find f^(-1)(x).

Transcript text: Given $f(x)=(x-2)^{3}+5$, find $f^{-1}(x)$.

Solution

Solution

Step 1: Express the function in terms of y

Given the function $f(x) = 1 \cdot (1 \cdot x - 2)^{\frac{1}{3}} + 5$, we aim to find its inverse.

Step 2: Isolate the root expression

First, isolate the root expression by subtracting 5 from both sides: $y - 5 = 1 \cdot (1 \cdot x - 2)^{\frac{1}{3}}$.

Step 3: Remove the coefficient of the root expression

Then, divide both sides by 1 to isolate the root expression: $\frac{y - 5}{1} = (1 \cdot x - 2)^{\frac{1}{3}}$.

Step 4: Eliminate the root

Raise both sides to the power of 3 to eliminate the root: $\left(\frac{y - 5}{1}\right)^{3} = 1 \cdot x - 2$.

Step 5: Isolate x

Finally, solve for x by subtracting -2 and dividing by 1: $x = \frac{\left(\frac{y - 5}{1}\right)^{3} + 2}{1}$.

$f^{-1}(x) = \frac{\left(\frac{x - 5}{1}\right)^{3} + 2}{1}$, rounded to 0 decimal places.

Was this solution helpful?

Unhelpful

Helpful