Question

Determine the rational function g(x) which is the inverse of the given function:

f(x) = x² - x - 6

Answer 1

`g(x)=\sqrt{x+6(1)/(4) } +(1)/(2)`

Explanation:

Turn the quadratic function into perfect quadratic function:

f(x) = x² - x - 6

divide the coefficient of x by 2 ⇒ coefficient of -x = -1

⇒ -1 ÷ 2 =
`-(1)/(2)`

⇒
`(x - (1)/(2) )^(2) =x^(2) -x+(1)/(4)`

⇒
`x^(2) -x-6=(x^(2) -x+(1)/(4) )-6(1)/(4)`

⇒
`x^(2) -x-6=(x-(1)/(2) )^2-6(1)/(4)`

`f(x)=(x-(1)/(2) )^2-6(1)/(4)`

`(x-(1)/(2) )^(2) =f(x)+6(1)/(4)`

`x-(1)/(2) =\sqrt{f(x)+6(1)/(4) }`

`x=\sqrt{f(x)+6(1)/(4) }+(1)/(2)`

now, change the x into g(x) & f(x) into x

`g(x)=\sqrt{x+6(1)/(4) } +(1)/(2)`