Suppose the graph of f is a parabola with vertex at (2,1). Suppose g(x) = 3 x + 2.
a) What are the coordinates of the vertex of the graph of g(composed)f?
b) What are the coordinates of the vertex of the graph of f(composed)g?
For hints on this kind of problems, see response to next post:
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To find the coordinates of the vertex of the graph of g(composed)f and f(composed)g, we need to understand what the composition of functions means. Let's break it down step-by-step:
a) To find the coordinates of the vertex of g(composed)f, we need to substitute the given function g(x) = 3x + 2 into x in the equation of f(x), which is a parabola with vertex at (2,1).
Step 1: Replacing x with g(x), we have:
f(g(x)) = f(3x + 2)
Step 2: To find the vertex, we need to find the value of x that gives us the vertex. In this case, we substitute the x-coordinate of the vertex of f, which is 2, into the composition function f(g(x)).
f(g(2)) = f(3(2) + 2) = f(8)
Since we don't have any information about the specific equation or expression for f(x), we can't determine the value of f(8). Therefore, we can't determine the exact coordinates of the vertex of g(composed)f without more information about function f.
b) To find the coordinates of the vertex of f(composed)g, we need to substitute the given function f(x) into g(x), which is a linear function g(x) = 3x + 2.
Step 1: Replacing x with f(x), we have:
f(g(x)) = f(3x + 2)
Step 2: We need to substitute the x-coordinate of the vertex of g, which is 2, into the composition function f(g(x)).
f(g(2)) = f(3(2) + 2) = f(8)
Similar to the previous case, without knowing the specific equation or expression for f(x), we can't determine the value of f(8). Therefore, we can't determine the exact coordinates of the vertex of f(composed)g without more information about function f.
To find the coordinates of the vertex of the composition functions, we need to evaluate the compositions and determine the vertex of each resulting graph.
a) To find the composition g(composed)f(x), we substitute f(x) into g(x):
g(composed)f(x) = g(f(x)) = 3(f(x)) + 2
Since the vertex of f is given as (2, 1), we substitute x = 2 into the function f(x) to find f(2) = 1.
Now we can substitute f(2) into g(composed)f(x):
g(composed)f(x) = 3(1) + 2 = 5
Therefore, the vertex of the graph of g(composed)f is (2, 5).
b) To find the composition f(composed)g(x), we substitute g(x) into f(x):
f(composed)g(x) = f(g(x))
Substituting g(x) = 3x + 2, we have:
f(composed)g(x) = f(3x + 2)
Since the vertex of f is given as (2, 1), we substitute x = 2 into the function f(x) to find f(2) = 1.
Now we can substitute 3x + 2 into f(composed)g(x):
f(composed)g(x) = f(3x + 2) = f(2) = 1
Therefore, the vertex of the graph of f(composed)g is (2, 1).
In summary:
a) The coordinates of the vertex of g(composed)f is (2, 5).
b) The coordinates of the vertex of f(composed)g is (2, 1).