Question

Find the inverse of the function f(x) = 2x² - 3x NO ABSURD ANSWERS IF YOU DON't WANT YOURSELVES TO GET REPORTED!

Answer 1

`\boxed{f^(-1)(x)= (√(8x+9)+3)/(4)}`

Explanation:

`f(x)=2x^2-3x`

`f(x)=y`

`y=2x^2-3x`

Switch variables.

`x=2y^2-3y`

Solve for y.

Multiply both sides by 8.

`8x=16y^2-24y`

Add 9 on both sides.

`8x+9=16y^2-24y+9`

Take the square root on both sides.

`√(8x+9) =√(16y^2-24y+9)`

Add 3 on both sides.

`√(8x+9)+3 =√(16y^2-24y+9)+3`

Divide both sides by 4.

`(√(8x+9)+3)/(4)= (√(16y^2-24y+9)+3)/(4)`

Simplify.

`(√(8x+9)+3)/(4)= (4y-3+3)/(4)`

`(√(8x+9)+3)/(4)= (4y)/(4)`

`(√(8x+9)+3)/(4)=y`

Inverse y =
`f^(-1)(x)`

`f^(-1)(x)= (√(8x+9)+3)/(4)`

Answer 2

`f^(-1)(x) = (3)/(4) \pm (1)/(4)√(8x + 9)`

Explanation:

`f^(-1)(x) = 2x^2 - 3x`

Change function notation to y.

`y = 2x^2 - 3x`

Switch x and y.

`x = 2y^2 - 3y`

Solve for y.

`2y^2 - 3y = x`

Complete the square on the left side. We must divide both sides by 2 to have y^2 as the leading term on the left side.

`y^2 - (3)/(2)y = (x)/(2)`

1/2 of 3/2 is 3/4. Square 3/4 to get 9/16.

Add 9/16 to both sides to complete the square.

`y^2 - (3)/(2)y + (9)/(16) = (x)/(2) + (9)/(16)`

Find common denominator on right side.

`(y - (3)/(4))^2 = (8x)/(16) + (9)/(16)`

If X^2 = k, then
`X = \pm √(k)`

`y - (3)/(4) = \pm \sqrt{(1)/(16)(8x + 9)}`

Simplify.

`y = (3)/(4) \pm (1)/(4)√(8x + 9)`

Back to function notation.

`f^(-1)(x) = (3)/(4) \pm (1)/(4)√(8x + 9)`