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Question 905689: Did I do this composite function correctly?
Let f(x) = 1/(2x), g(x) = 7x^3, and h(x) = −4x^2 + 1
(f º g º h)(x) = 1/(686(-4x^2+1)^3)
Is that right?
Thank You
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i don't believe so.
i got something entirely different and i confirmed it is correct.
start with:
f(x) = 1/(2x)
g(x) = 7x^3
h(x) = -4x^2 + 1
(f.g.h)(x) = f(g(h(x)))
that translates to f of g of h of x.
you find h(x) first.
then you find g(h(x)) next.
then you find f(g(h(x))) next.
h(x) = (-4x^2 + 1)
g(h(x)) = g(-4x^2 + 1) = (7 * (-4x^2 + 1)^3)
f(g(h(x)) = f(7 * (-4x^2 + 1)^3) = 1 / (2 * (7 * (-4x^2 + 1)^3))
if you work it all through, you will get:
f(g(h(x)) = 1 / (-896x^6 + 672x^4 - 168x^2 + 2)
the piece parts are:
h(x) = -4x^2 + 1
g(h(x)) = -448x^6 + 336x^4 - 84x^2 + 1
f(g(h(x)) = 1 / (-896x^6 + 672x^4 - 168x^2 + 2)
i confirmed by assigning the arbitrary value of 7 to x and then solving from the original equations and then solving from the final equation.
i got:
h(7) = -195
g(-195) = -51904125
f(-51904125) = -9.63314573*10^-9
I then went to the final equation to get:
(f.g.h)(7) = f(g(h(7)) = -9.63314573*10^-9
since both answers are the same, i assume that i did it right.
you can find (-4x^2 + 1)^3 by first multiplying (-4x^2 + 1) by (-4x^2 + 1) and the multiplying the result by (-4x^2 + 1).
you can also find it by seeing that it is a binomial expansion of (a + b)^3 where a = -4x^2 and b = 1.
i did it both ways and got the same answer of (-4x^2 + 1)^3 = -64x^6 + 48x^4 - 12x^2 + 1
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