Find f o g and g o f, if they exist. State the domain and range
f={(-8,-4),(0,4),(2,6),(-6,-2)}
g={(4,-4),(-2,-1),(-4,0)(6,-5)}
To find f o g (f composed with g), we need to substitute the output of g into f.
First, let's determine the domain and range of f and g:
Domain of f = {-8, 0, 2, -6} (the x-values of f)
Range of f = {-4, 4, 6, -2} (the y-values of f)
Domain of g = {4, -2, -4, 6} (the x-values of g)
Range of g = {-4, -1, 0, -5} (the y-values of g)
Now, let's compute f o g:
For each (x, y) pair in g, we substitute the output y into f.
For g = (4, -4):
f(g) = f(-4) = -2
For g = (-2, -1):
f(g) = f(-1) = Not defined in f
For g = (-4, 0):
f(g) = f(0) = 4
For g = (6, -5):
f(g) = f(-5) = Not defined in f
So, f o g = {(-4, -2), (0, 4)}
The domain of f o g is the set of x-values from g, which is {4, -2}.
The range of f o g is the set of y-values from f after applying g, which is {-2, 4}.
Now, let's compute g o f:
For each (x, y) pair in f, we substitute the output y into g.
For f = (-8, -4):
g(f) = g(-4) = Not defined in g
For f = (0, 4):
g(f) = g(4) = -4
For f = (2, 6):
g(f) = g(6) = -5
For f = (-6, -2):
g(f) = g(-2) = -1
So, g o f = {(0, -4), (2, -5), (-6, -1)}
The domain of g o f is the set of x-values from f, which is {-8, 0, 2, -6}.
The range of g o f is the set of y-values from g after applying f, which is {-4, -5, -1}.