Answer:
- f(x) = x^5
- g(x) = 2x -6 . . . . . and see below for other possible definitions
Step-by-step explanation:
You are given an expression for the composition H(x) = f(g(x)) and asked to decompose it into two functions. One way to do that is to look at what is being done to the variable in the function H(x):
- the variable is multiplied by 2
- 6 is subtracted from the product
- the difference is raised to the 5th power.
One of the ways to create the functions g(x) and f(x) is to start at the top of this list and work your way down. Any subset of these transformations can be made into g(x). Then the rest of them are made into f(x).
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For example, using
g(x) = 2x
Then the rest of the list is f(x):
f(x) = (x-6)^5
so when you put 2x as the argument for f(x), you get ...
H(x) = f(g(x)) = f(2x) = (2x -6)^5 . . . . . the function we want.
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We could also do the first two steps of our list as g(x):
g(x) = 2x -6
f(x) = x^5
so
H(x) = f(g(x)) = f(2x -6) = (2x -6)^5
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If we like, we can factor the expression for H(x):
H(x) = (2(x -3))^5 = 32(x -3)^5
Using the methods above, we could write ...
g(x) = x -3
f(x) = 32x^5
so
H(x) = f(g(x)) = f(x -3) = 32(x -3)^5
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The multiplication and the addition can be made into several parts. We could choose ...
g(x) = (1/4)x -1
f(x) = (8x +2)^5
so
H(x) = f(g(x)) = f(1/4x -1) = (8(1/4x -1) +2)^5 = (2x -6)^5
