Map IT!
So Liam says the formations f(x)=x² and g(x)=(–x)² are the same... but are they? Let's explore mappings to figure it out!
These sure do look the same . . .
SOLDIERS
One way to investigate is to look at where each order sends our soldiers.

f(x) sends soldier 1 to f(1)=1²=1 and g(x) sends soldier 1 to g(1)=(1)²=1.

f(x) sends soldier 2 to f(2)=2²=4 and g(x) sends soldier 2 to g(2)=(2)²=4.
It looks like both orders, f(x) and g(x), send our soldiers to the same location! Since
(x)²=(x)·(x)=(1)(x)(1)(x)=(1·1)(x·x)=x²
this means f(x) and g(x) send every soldier to the same location. Maybe Liam is right?
BUS ROUTES
Mr. Pikake tells the group to "think of your mappings!". He is referring to the bus routes the students studied way back in chapter 1. To obtain g(x)=(–x)² we must use a transformation which we can express as a combination of orders. Since f(x)=x², let k(x)=–x, then f(k(x))=(–x)²=g(x). What would this route look like?
and
then
First, each soldier is sent to the station of their evil twin. This creates a line, a march, with a slope of –1.
Next, you can think of a machine that transports every soldier from their first assigned location. The machine moves them to the appropriate location given by the second order.
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The mapping, or bus route, for this function, g(x)=(–x)², takes 1→–1→1.
The route for f(x)=x² takes 1→1 (no stop!).
So while the ultimate formation is the same, the route the soldiers have taken to get there is different.
ANOTHER TRICKY FORMATION
Here's another deceiving formation!
Kernel 1 created a command. She started with the formation f(x)=x³ and reflected it over center stage to find g(x)=(–x)³. Finally, she had everyone take one step forward making her ending formation k(x)=(–x)³+1.
Kernel 8 had another idea. He started with the formation a(x)=x³ and instead reflected it over ground zero to find b(x)=–x³. Then he had everyone take one step forward to form c(x)=–x³+1.
Are these formations the same?
Explore these orders!
©MathBait created with GeoGebra
WAS LIAM RIGHT?!
Yes and No! Liam is right about the resulting formations. If we only look at where every soldier ends up, the formations are identical. But, if we care about the journey, then the commands are very different.
We also must be careful and consider our dominion. Transforming a formation can impact the soldiers allowed in! This means while two orders may look the same, they might be incredibly different!
Consider the following:
Are these formations the same?
and
These are not the same orders! While we might have been told we can just "cross out" the factors of x3 in g(x), this is not true! To simplify g(x), it must be that (x3)/(x3)=1. This is true, but only if x≠3! That means, a commander cannot use the soldier x=3 in the order g(x). Because of this, they end up with a big hole hole in the march.
CONCLUSION
Just because two things look the same, it doesn't mean they are the same!