Question

The question says Draw the reflection of f(x)=sqrt x-2 in the line y=x which I done but now I need to write the equation of the reflection in the form of y=g(x). I'm not sure how to write this equation using this form. Could you please help. Thanks

Answer 1

The reflection of a function about the line y=x is the inverse.

To find the inverse of f(x)=sqrt(x-2), we proceed as follows:
1. let y=f(x)
2. interchange the variables x and y.
3. solve for the new variable y in terms of x.

1. y=sqrt(x-2)
2. x=sqrt(y-2)
3. x²=y-2
g(x)=y=x²+2

g(x) is the inverse of f(x), and it is also the reflection of f(x) about y=x.

Answer 2

by reflecting a relation in y = x is really the same as taking the inverse of the relation

so the inverse of y = √(x-2) is
x = √(y-2)
then x^2 = y - 2
y = x^2 + 2 , of course only valid for x ≥ 2

(did you notice that the steps to calculate a value for the second equation is in the reverse order using the inverse operations?
e.g.
for the original let x = 6
first we subtract 2 then take the square root to get 2
for the inverse relations y = x^2 +2
the first thing we do is square the number, then add 2 , or 2^2 + 2 = 6, the original value we started with )