Question 905689
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