To find the inverse of a function, we swap every x-coordinate for a y-coordinate, and likewise. This is because inverse functions are reflected over the line y=x, which has the reflection rule: (x, y)——>(y,x). So, the x and y coordinates flip positions, and this creates an inverse function by definition. Now, once we invert our x and y-coordinates, we must solve for y, since the inverse function is still a function. We will lastly replace y with y^-1, indicating an inverse function.
1.) Swap all x-coordinates for y-coordinates and all y-coordinates for x-coordinates.
Given the function: g(x)=(5th root)√(2x-1)-1
• Replace g(x) with y, since y=g(x):
(y)=(5th root)√(2x-1)-1
• Now, swap all x and y-coordinates:
(x)=(5th root)√(2(y)-1)-1
• Solve for y:
• Add 1 to both sides:
x+1=(5th root)√(2y-1)
• Because 2y-1 is inside the radical, we will take the radical to the 5th power to cancel out the 5th root. We must take each side to the 5th root to maintain equality.
(x+1)^5=[(5th root)√(2y-1)-1]^5
(x+1)^5=2y-1
• Add 1 to both sides:
((x+1)^5)+1=2y
• Divide by 2 on both sides:
[((x+1)^5)+1]/2=2y/2
[((x+1)^5)+1]/2=y
• Apply the symmetric property: if a=b, then b=a
y=[((x+1)^5)+1]/2
• Lastly, replace “y” with g^-1 to indicate an inverse function of g(x).
g^-1= [((x+1)^5)+1]/2
*The brackets are used to show the entire right side is divided by 2, and parentheses are used to group terms together.
Hope this helps!