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

1 Answer

5 votes

Answer:

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)

→ Divide both sides by 3


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

User AFF
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