inverse of the 5th root of (3x-1)/(x-2)

Question

I need to find the inverse of the 5th root of (3x-1)/(x-2)

Answer 1

Hi Lynn,

 

This graph helps to show what is going on:

 

https://www.desmos.com/calculator/kcmsugldks

 

One method for explicitly defining the inverse function is to "interchange x and y), and then solve for y.

 

So:

 

x = [(3y-1)/(y-2)]^1/5

 

To solve that for y:

1. Raise both sides to the 5th power.

 

x⁵ = (3y-1)/(y-2)

 

Multiply both sides by (y-2)

 

x⁵y-2x⁵ = 3y -1

 

Place terms containing y on left side and other terms on right side:

 

x⁵y - 3y = 2x⁵ - 1

 

Factor y from the left side:

 

y(x⁵ - 3) = 2x⁵ - 1

 

Divide both sides by (x⁵ -3) giving:

 

y = (2x⁵ - 1)/(x⁵ - 3)

 

There is at least one other consideration--does the original function have an inverse "function", i.e. does it pass the horizontal line test? The graph seems to confirm that in fact it does.

 

If this is from a Calculus class you could apply the first derivative test to show that the function is always decreasing and therefore does have an inverse function without having to limit its domain.

 

By the way if you visit the graph you can hide a graph by clicking its icon in the left panel.