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
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.
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 )
Sure, I'd be happy to help you write the equation of the reflection of the function f(x) = √(x-2) in the line y = x in the form of y = g(x).
To find the reflection of a function in the line y = x, we need to swap the x and y coordinates of each point on the original function. In other words, if we have a point (x, y) on the original function, the reflected point will be (y, x).
Let's go through the steps:
1. Start with the original function f(x) = √(x-2).
2. Swap the x and y coordinates of each point:
- Substitute x with y and y with x in the original function:
x = √(y - 2)
3. Now solve for y to get the equation in the form of y = g(x).
- Square both sides of the equation to eliminate the square root:
x^2 = y - 2
- Add 2 to both sides to isolate y:
x^2 + 2 = y
So, the equation of the reflection of f(x) = √(x-2) in the line y = x can be written in the form of y = g(x) as:
g(x) = x^2 + 2
I hope this helps! If you have any more questions, feel free to ask.