Question

How do you find the inverse of `y =sqrt(x-4)`?

Answer 1

`x=y^2+4`
or if you wish to keep `y` as the dependent variable:
`y=x^2+4`

Explanation:

Method 1:
`y=sqrt(x-4)`

`rArr y^2 = x-4`

`rArr y^2+4 = x color(white)("XXx")orcolor(white)("XXX")x=y^2+4`

Method 2: (perhaps more complex but more technically accurate)
Let
`color(white)("XXX")f(x)=y`
and
`color(white)("XXX")barf (x)` be the inverse of `f(x)`

By definition of inverse:
`color(white)("XXX")f(barf(x)) = x`

But since
`color(white)("XXX")f(x) = sqrt(x-4)`
`rArr`
`color(white)("XXX")f(barf(x))= sqrt(barf(x)-4)`

So
`color(white)("XXX")sqrt(barf(x)-4) = x`

`color(white)("XXX")barf(x)-4 = x^2`

`color(white)("XXX")barf(x)=x^2+4`