Expert: Sherman D. - 9/7/2005

Question
if u(x)=4x-5, v(x)=x^2, and f(x)=1/x, find formulas for the following:

1) u(v(f(x)))
2) v(u(f(x)))
3) f(u(v(x)))


Answer
u(x) = 4x - 5
v(x) = x^2
f(x) = (1/x)

1.)
u(v(f(x)))
f(x) = (1/x)
v(1/x) = (1/x)^2 = (1/(x^2))
u(1/(x^2)) = 4(1/(x^2)) - 5 = (4/(x^2)) - 5

so u(v(f(x))) = (4/(x^2)) - 5
or you can probably write it this way
u(v(f(x))) = (2/x)^2 - 5 or (-5x^2 + 4)/(x^2)

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2.)
v(u(f(x)))
f(x) = (1/x)
u(1/x) = 4(1/x) - 5 = (4/x) - 5
v((4/x) - 5) = ((4/x) - 5)^2
v((4/x) - 5) = ((4/x) - 5)((4/x) - 5)
v((4/x) - 5) = (16/(x^2)) - (20/x) - (20/x) + 25
v((4/x) - 5) = (16/(x^2)) - (40/x) + 25

so v(u(f(x))) = (16/(x^2)) - (40/x) + 25
or you can probably write it like this
v(u(f(x))) = (4/x)^2 - (40/x) + 25
or
(25x^2 - 40x + 16)/(x^2)
or
((5x - 4)/x)^2

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3.)
f(u(v(x)))
v(x) = x^2
u(x^2) = 4x^2 - 5
f(4x^2 - 5) = 1/(4x^2 - 5)

so f(u(v(x))) = 1/(4x^2 - 5)