Find the inverse function when the inverse of f(-6)

Question

The function f(x) = (2x)^(1/3)−7 is invertible, find f^−1(−6).

Answer 1

The steps to find the inverse are:

1. Replace f(x) with y: y = (2x)^(1/3)−7
2. Switch the x and y variables: x = (2y)^(1/3)−7
3. Solve for y:

Add 7 to both sides and simplify

x+7 = (2y)^(1/3)−7+7

x+7 = (2y)^(1/3)

Cube both sides and simplify

(x+7)³ = [(2y)^(1/3)]³

(x+7)³=2y

Divide both sides by 2 and simplify

(x+7)³/2=2y/2

(x+7)³/2=y

Replace y with f^-1(x)

f^-1(x)=(x+7)³/2

Now this can be evaluated at -6 by plugging it in an simplifying

f^-1(-6)=(-6+7)³/2

=(1)³/2

= 1/2

Answer 2

f(x) = (2x)^1/3 - 7

The domain of f^-1(x) is the range of f(x).

To find f^-1(-6), find the value of x so that f(x) = -6.

So, (2x)^1/3 - 7 = -6

(2x)^1/3 = 1. So, 2x = 1 . x = 1/2