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Find a formula for the inverse of the function.
f(x) = 3 + √(5+6x).

Sagot :

Answer:

Find a formula for the inverse of the function. f(x)= 3+ 5+6x f-1(x)=

Rvkacademicidn

Answer:

The inverse of f(x) is

[tex]f^{-1}(x) =\boxed{ \dfrac{x^2 -6x + 4 }{6}}[/tex]

Step-by-step explanation:

The given function is

[tex]f(x) = 3 + \sqrt{5 + 6x[/tex]

and we are asked to find the inverse

Proceed as follows:

  • Let y = f(x). Then we get the function as
    [tex]y= 3 + \sqrt{5 + 6x[/tex]
  • Interchange x and y in the above function:
    [tex]x = 3 + \sqrt{5 + 6y}[/tex]
  • Solve the equation for y:
    [tex]x - 3 = \sqrt{5 + 6y} \quad $ (subtract 3 from both sides)\\\\(x - 3)^2 = 5 + 6y \quad $ (square both sides)\\\\(x-3)^2 - 5 = 6y \quad $ (Subtract 5 from both sides)\\\\6y = (x-3)^2 - 5 \quad $ (switch sides)\\\\6y = (x^2 - 6x + 9) - 5 \quad $ (expand (x-3)^2\\\\6y = x^2 -6x + 4 \quad $ (simplify right side)\\\\y = \dfrac{x^2 -6x + 4 }{6} \quad $ (Divide by 6 both sides)[/tex]

The right side represents the inverse of the function f(x)

Inverse:
[tex]f^{-1}(x) = \dfrac{x^2 -6x + 4 }{6}[/tex]

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