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

In summary: So it is not necessarily a general rule but something that you would need to specifically think about for each function.
  • #1
madah12
326
1

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?
 
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  • #2
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?
 
  • #3
yes the positive one ,so is this a general rule for roots in inverse function?
 
  • #4
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.
 
  • #5
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, [itex]y= 2x^2- 8x[/itex] 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,
[itex]y= 2x^2- 8x[/itex] for [itex]x\le 2[/itex] and [itex]y= 2x^2- 8x[/itex] for [itex]x\ge 2[/itex]. Because your problem said "[itex]x\ge 2[/itex]" you are talking about the second.

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

How do I know to take the positive root? Because [itex]x\ge 2[/itex], after "swapping" x and y, became [itex]y\ge 2[/itex]. It is the positive root that makes y greater than or equal to 2. Taking the negative root, y would be "2 minus something".
 
  • #6
HallsofIvy said:
How do I know to take the positive root? Because [itex]x\ge 2[/itex], after "swapping" x and y, became [itex]y\ge 2[/itex]. 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.
 

1. What is the process for finding the inverse of a function?

The process for finding the inverse of a function involves swapping the x and y variables and solving for y. This will give you the inverse function, which can be used to find the input value (x) when given the output value (y).

2. How do I know which sign of the root to take when finding the inverse of a function?

The sign of the root to take when finding the inverse of a function depends on the original function. If the original function has an even power (e.g. x^2), then the inverse function will have a positive and negative root. If the original function has an odd power (e.g. x^3), then the inverse function will have only one root.

3. Can the sign of the root change when finding the inverse of a function?

Yes, the sign of the root can change when finding the inverse of a function. This depends on the original function and whether it has an even or odd power. If the original function has an even power, then the inverse function will have both positive and negative roots. If the original function has an odd power, then the inverse function will have only one root.

4. What happens if the original function has multiple roots?

If the original function has multiple roots (e.g. x^2 - 4 = 0 has roots of -2 and 2), then the inverse function will also have multiple roots. This means that there will be more than one input value (x) that corresponds to a given output value (y).

5. Is there a way to check if I have correctly found the inverse of a function?

Yes, there are two ways to check if you have correctly found the inverse of a function. First, you can plug in the inverse function into the original function and see if you get back the original input value. Second, you can graph both the original function and the inverse function and see if they are reflections of each other over the line y = x.

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