Answer:
The answer is below
Step-by-step explanation:
Show that f(x) f(y) = f(x+y)
From trigonometric:
sin(x + y) = sinxcosy + cosxsiny
sin(x - y) = sinxcosy - cosxsiny
cos(x + y) = cosxcosy - sinxsiny
cos(x - y) = cosxcosy + sinxsiny
![f(x)=\left[\begin{array}{ccc}cosx&-sinx&0\\sinx&cosx&0\\0&0&1\end{array}\right] ,f(y)=\left[\begin{array}{ccc}cosy&-siny&0\\siny&cosy&0\\0&0&1\end{array}\right] \\\\\\f(x)f(y)=\left[\begin{array}{ccc}cosxcosy-sinxsiny&-cosxsiny-sinxcosy&0\\sinxcosy+cosxsiny&-sinxsiny+cosxcosy&0\\0&0&1\end{array}\right] \\\\\\f(x)f(y)=\left[\begin{array}{ccc}cos(x+y)&-sin(x+y)&0\\sin(x+y)&cos(x+y)&0\\0&0&1\end{array}\right] \\\\\\](https://tex.z-dn.net/?f=f%28x%29%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dcosx%26-sinx%260%5C%5Csinx%26cosx%260%5C%5C0%260%261%5Cend%7Barray%7D%5Cright%5D%20%2Cf%28y%29%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dcosy%26-siny%260%5C%5Csiny%26cosy%260%5C%5C0%260%261%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%5C%5C%5C%5Cf%28x%29f%28y%29%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dcosxcosy-sinxsiny%26-cosxsiny-sinxcosy%260%5C%5Csinxcosy%2Bcosxsiny%26-sinxsiny%2Bcosxcosy%260%5C%5C0%260%261%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%5C%5C%5C%5Cf%28x%29f%28y%29%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dcos%28x%2By%29%26-sin%28x%2By%29%260%5C%5Csin%28x%2By%29%26cos%28x%2By%29%260%5C%5C0%260%261%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%5C%5C%5C%5C)
![f(x+y)=\left[\begin{array}{ccc}cos(x+y)&-sin(x+y)&0\\sin(x+y)&cos(x+y)&0\\0&0&1\end{array}\right] \\\\\\Therefore\ f(x)f(y)=f(x+y)](https://tex.z-dn.net/?f=f%28x%2By%29%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dcos%28x%2By%29%26-sin%28x%2By%29%260%5C%5Csin%28x%2By%29%26cos%28x%2By%29%260%5C%5C0%260%261%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%5C%5C%5C%5CTherefore%5C%20f%28x%29f%28y%29%3Df%28x%2By%29)
Answer:
Hello there! I hope your having a good day. I'm not exactly sure about this answer: 50o=30 u+(20m+.45a)
Step-by-step explanation:
Let g the inverse function of f.
The most important property of g and f being inverses of each other, is that
g(f(x))=x, also f(g(x))=x
so, what one function 'does' to x, the other 'undoes' it.
Thus, we have:
f(g(x))=x and alos f(g(x))= -g(x)+3, from the rule
thus :
-g(x)+3=x
-g(x)=x-3
g(x)=-x+3
check: f(g(x))=f(-x+3)=-(-x+3)+3=x-3+3=x
Answer: the inverse of f is g, such that g(x)=-x+3
