Can you check my calculations please
g maps points (0,0),(1,0),(0,1) to (6,-2),(6,-1),(5,-2)
So, g(x) = (0 -1) .....(6)
..................(1 0)x + (-2)
f is a reflection through y=x-2
so, f(x) = (0 1) .....(2)
...............(1 0)x + (-2)
So; f o g = g(f(x))
= (0 -1)(0 1) ...(0 -1)(2) .....(6)
...(1 0)(1 0)x + (1 0)(-2) + (-2)
= (-1 0) ....(2) ....(6)
....(0 1)x + (2) + (-2)
= (-1 0) .....(8)
....(0 1)x + (0)
This is really stretching my brain.
Sure, let's go through your calculations step by step and check if they are correct.
You have given the mapping g that maps points (0,0), (1,0), and (0,1) to (6,-2), (6,-1), and (5,-2) respectively. To represent this mapping as a matrix, you wrote g(x) = (0 -1) (6) (1 0)x + (-2). This matrix representation is correct.
Next, you defined a reflection mapping f through the line y = x - 2. To represent this mapping as a matrix, you wrote f(x) = (0 1) (2) (1 0)x + (-2). This matrix representation is also correct.
Now, you want to calculate the composition of f and g, which is denoted as f o g. To do this, you need to multiply the matrices of f and g in the correct order.
When you multiplied the matrices (0 -1)(0 1) and (0 -1)(2), you obtained the matrix (-1 0) (2) (6). This multiplication step is correct.
However, in the next step, you made an error. Instead of multiplying (1 0)(1 0)x and (1 0)(-2) + (-2), you should have multiplied (0 1)x and (2) + (-2). The correct equation should be:
(-1 0) (2) (6)
(0 1)x + (0)
After performing this multiplication correctly, you would get the following matrix:
(-1 0) (8)
(0 1)x + 0
So, your final result for f o g is (-1 0) (8) (0 1)x + 0.
Remember, when performing matrix multiplications, it is crucial to ensure that the dimensions of the matrices match and follow the appropriate order. Double-checking your steps can help identify and correct any mistakes. Keep practicing, and don't worry, understanding matrix compositions can be challenging, but with practice, you'll get the hang of it.