Question

Let f:[0, infinity) to R be a function defined by f(x)= 9x² +6x-5. Proove that f is not invertible.Modify ,only the codomain of f to make f invertible and then find it's inverse .Also , find f^{​​​​​​-1}(43)

Answer 1

Let y = f(x) = 9x^2 + 6x - 5

= 9x^2 + 6x + 1 - 6

= (3x + 1)^2 - 6

=> y + 6 = (3x + 1)^2

=> √(y + 6) = 3x + 1

=> 3x = √(y + 6) - 1

=> x = {√(y + 6) - 1}/3

For x to be defined, we have

y + 6 ≥ 0

=> y ≥ -6

But given codomain = R

Hence, the given function is not invertible.

It will be invertible when the function is

f : [0, ∞) --> [-6, ∞)

Again,

Let y = f(x) = 9x^2 + 6x - 5

= 9x^2 + 6x + 1 - 6

= (3x + 1)^2 - 6

=> y + 6 = (3x + 1)^2

=> √(y + 6) = 3x + 1

=> 3x = √(y + 6) - 1

=> x = {√(y + 6) - 1}/3

=> f^-1 (x) = {√(x + 6) - 1}/3

So, the inverse of the function is {√(x + 6) - 1}/3

Now to find f^-1(43) put x = 43 in the above equation and get the answer.
Thank you!