How to know which sign of root to take when finding the inverse of a function

[madah12]

Homework Statement

find the inverse of
y=2x^2-8x
x>=2
I am in calculus b but forgot some of the algebra so I am not sure how to treat the root

Homework Equations

The Attempt at a Solution

I am skipping steps

y=2(x-2)^2-8

(y+8)/2=(x-2)^2
I know that x >=2
so f(2) =-8 and it is increasing so the range is [-8,infinity)
now how can I know which root to take?

 

[Mentallic]

As [itex]x\to \infty[/tex], [itex]y\to \infty[/tex] so when you find the inverse, this will still apply since they are images about the y=x line. Now can you see which root you should be taking?[/itex][/itex]

 

[madah12]

yes the positive one ,so is this a general rule for roots in inverse function?

 

[Mentallic]

Umm no, not necessarily. I just thought of it logically for the inverse of quadratics, so I haven't extended the same idea to other functions.

The best way would be to draw a rough sketch of the quadratic, then draw its inverse by flipping it about the line y=x and go from there. Pictures can really help.

 

[HallsofIvy]

What is true here is that, strictly speaking, f does NOT HAVE an inverse. In order to talk about an inverse for a function that is not one-to-one, you have to divide the domain into pieces, taking a different inverse for each "piece".

Here, $y= 2x^2- 8x$ and it is easy to see that its graph is a parabola, opening upward with vertex at (2, -8). We would need to divide this into two functions,
$y= 2x^2- 8x$ for $x\le 2$ and $y= 2x^2- 8x$ for $x\ge 2$. Because your problem said "$x\ge 2$" you are talking about the second.

To find the inverse, swap x and y to get $x= 2y^2- 8y$ and solve for y, perhaps by completing the square:
$$x= 2(y^2- 4y)= 2(y^2- 4y+ 4- 4)= 2(y- 2)^2- 8$$
$$2(y- 2)^2= x+ 8$$
$$(y- 2)^2= \frac{x+ 8}{2}$$
$$y- 2= \sqrt{\frac{x+8}{2}}$$
$$y= 2+ \sqrt{\frac{x+8}{2}}$$

How do I know to take the positive root? Because $x\ge 2$, after "swapping" x and y, became $y\ge 2$. It is the positive root that makes y greater than or equal to 2. Taking the negative root, y would be "2 minus something".

 

[Mentallic]

How do I know to take the positive root? Because $x\ge 2$, after "swapping" x and y, became $y\ge 2$. It is the positive root that makes y greater than or equal to 2. Taking the negative root, y would be "2 minus something".

Ahh good point.