whats the inverse of the function of y= 3x2 -2x +1

Question

3x² -2x +1

Answer 1

Nece,

 

One method is to put the expression in the form y = a(x-h)² + k by completing the square and then solve for x.

 

y = 3x² - 2x + 1

y = 3[x² - (2/3)x + (1/3)]

y = 3[x² - (2/3)x + (1/9)] + [1 - (1/3)]

y = 3[x-(1/3)]² + 2/3

 

Subtract 2/3 from both sides and then divide by 3 to get;

 

[y-(2/3)]/3 = [x-(1/3)]²

 

or [(y/3) - (2/9)] = [x-(1/3)]²

 

This can also be written as:

[(3y/9) - (2/9)] = [x-(1/3)]²

 

Take square root of both sides and solve for x:

 

x = +/- [(3y/9-(2/9)]^1/2 + 1/3   = +/-(1/3)[3y-2]^1/2 + 1/3

 

Now interchange x and y:

 

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

 

Note that the domain for the inverse function is x>= 2/3.

 

Hope this helps!

 

Answer 2

To find an inverse, you swap your x's and y's, and then solve for the new y.  You will need to "complete the square" for this problem to deal with the multiple variable pieces:

 

• x=3y²-2y+1
• x=3(y²-2y/3)+1
• x=3(y-2y/3+1/3)+1-1/3 ~ adding 1/3 to complete the square, and then subtracting it to maintain equality
• x=3(y-1/3)²+2/3
• (x-2/3)/3=(y-1/3)²
• (3x-2)/9=(y-1/3)² ~ algebraic clean up
• y-1/3=+/- root(3x-2)/3
• y=+/- root(3x-2)/3+1/3