Question

Find the inverse when f(x)=3x^2-3x-2

Answer 1

The inverse of f(x) is
`f^(-1)`(x) = ±
`\sqrt{(x+(11)/(4))/(3)}` +
`(1)/(2)`

Explanation:

To find the inverse of the quadratic function f(x) = ax² + bx + c, you should put it in the vertex form f(x) = a(x - h)² + k, where

• h =
  `(-b)/(2a)`

• k is the vlue f at x = h

∵ f(x) = 3x² - 3x - 2

→ Compare it with the 1st form above to find a and b

∴ a = 3 and b = -3

→ Use the rule of h to find it

∵ h =
`(-(-3))/(2(3))` =
`(3)/(6)` =
`(1)/(2)`

∴ h =
`(1)/(2)`

→ Substitute x by the value of h in f to find k

∵ k = 3(
`(1)/(2)`)² - 3(
`(1)/(2)`) - 2

∴ k =
`-(11)/(4)`

→ Substitute the values of a, h, and k in the vertex form above

∵ f(x) = 3(x -
`(1)/(2)`)² +
`-(11)/(4)`

∴ f(x) = 3(x -
`(1)/(2)`)² -
`(11)/(4)`

Now let us find the inverse of f(x)

∵ f(x) = y

∴ y = 3(x -
`(1)/(2)`)² -
`(11)/(4)`

→ Switch x and y

∵ x = 3(y -
`(1)/(2)`)² -
`(11)/(4)`

→ Add
`(11)/(4)` to both sides

∴ x +
`(11)/(4)` = 3(y -
`(1)/(2)`)²

→ Divide both sides by 3

∵
`(x+(11)/(4))/(3)` = (y -
`(1)/(2)`)²

→ Take √ for both sides

∴ ±
`\sqrt{(x+(11)/(4))/(3)}` = y -
`(1)/(2)`

→ Add
`(1)/(2)` to both sides

∴ ±
`\sqrt{(x+(11)/(4))/(3)}` +
`(1)/(2)` = y

→ Replace y by
`f^(-1)`(x)

∴
`f^(-1)`(x) = ±
`\sqrt{(x+(11)/(4))/(3)}` +
`(1)/(2)`

∴ The inverse of f(x) is
`f^(-1)`(x) = ±
`\sqrt{(x+(11)/(4))/(3)}` +
`(1)/(2)`