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]
