Flipping Functions: Determining Vertical/Horizontal Flips

[soulmartyr]

The problem shows two graphs, the first of which is just f(x)=3√x (square root of x cubed -not sure how to make it look like that-) and it wants me to give a function for the graph on the right.
The second graph is flipped, vertically shifted 1 up f(x)+ 1 and horizontally shifted 2 right f(x-2).
My problem is/was I can't determine if it is a vertical or horizontal flip because square root of x cubed looks the same both ways. So I have an assumption, I would like to know if it's correct and a question.

It would have to be a vertical flip because a negative can be on the outside of a square root but not inside and still be a function?

Also if I come across this problem again, and the horizontal and vertical flip would both look the same what are some better ways of determining which it is?

 

[Nathanael]

The problem shows two graphs, the first of which is just f(x)=3√x (square root of x cubed -not sure how to make it look like that-)

The square root of x cubed looks like this $\sqrt{x^3}$ but I'm assuming what you meant to say is the cube root of x? $\sqrt[3]{x}$ Like that?

It would have to be a vertical flip because a negative can be on the outside of a square root but not inside and still be a function?

Not true. Since it's a cube root, you can have a negative inside of it with no complications.
For example, the cube root of negative 8 is negative two, and the cube root of positive 8 is positive 2.

My problem is/was I can't determine if it is a vertical or horizontal flip because square root of x cubed looks the same both ways.

It looks the same, because it IS the same. There is no difference between a horizontal and vertical flip (in this case)

To flip it horizontally, you replace x with negative x (because you want to 'flip' the x axis) and so you get $f(x)=\sqrt[3]{-x}$

To flip it vertically you replace y (or "f(x)") with negative y (because you want to 'flip' the y axis) so you get $f(x)=-\sqrt[3]{x}$


Let's compare those two functions.
Not only do they look the same visually (which is what is confusing you) but they ARE the same.

Look:

$\sqrt[3]{-x}=\sqrt[3]{(-1)(x)}=\sqrt[3]{-1}\sqrt[3]{x}=-1\sqrt[3]{x}=-\sqrt[3]{x}$



So you see, it truly does not matter which way you do it. Not visually, and not mathematically.

 

[Mark44]

The problem shows two graphs, the first of which is just f(x)=3√x (square root of x cubed -not sure how to make it look like that-)

This is very confusing. You wrote 3√x, which is 3 times the square root of x. You described this as the square root of x cubed, which could be either this --
$$\sqrt{x^3}$$
or this --
$$(\sqrt{x})^3$$

How you wrote it makes me think that you meant the cube root of x, which is $\sqrt[3]{x}$. Which one is your problem?

and it wants me to give a function for the graph on the right.
The second graph is flipped, vertically shifted 1 up f(x)+ 1 and horizontally shifted 2 right f(x-2). My problem is/was I can't determine if it is a vertical or horizontal flip because square root of x cubed looks the same both ways. So I have an assumption, I would like to know if it's correct and a question.

It would have to be a vertical flip because a negative can be on the outside of a square root but not inside and still be a function?

Also if I come across this problem again, and the horizontal and vertical flip would both look the same what are some better ways of determining which it is?

 

[Nathanael]

How you wrote it makes me think that you meant the cube root of x, which is $\sqrt[3]{x}$. Which one is your problem?

What makes me even more certain that they meant $\sqrt[3]{x}$ is that they said (or implied) that the function is odd.

 

[soulmartyr]

The square root of x cubed looks like this $\sqrt{x^3}$ but I'm assuming what you meant to say is the cube root of x? $\sqrt[3]{x}$ Like that?

I did mean cube root of X
ty and sorry for the confusion

your comparison was very helpful ty

Where do I learn how to make the rest of the symbols, besides the 'Quick symbols' given

 

[Nathanael]

I did mean cube root of X
ty and sorry for the confusion

your comparison was very helpful ty

Where do I learn how to make the rest of the symbols, besides the 'Quick symbols' given

If you look at the top on the very right there's a sigma symbol $\Sigma$ which has a lot more symbols (it can sometimes be annoying to find what you're looking for at first)

A lot of them are simple though, and so if you use them enough you'll just type it out by hand

(for example, "x^2" gives you $x^2$)

 

[soulmartyr]

awesome thanks again