Let # f(x)=ln(0.5x) # and #g(x)=e^(3x)#, how do you find each of the compositions?

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Answer 1 · 2015-07-28T00:53:13

Function composition is described as follows:

#f(g(x)) = f(u)# such that #u = g(x)#

Let's say you had instead:

#f(u) = 2u#
#g(x) = x^2#

Then:
#f(u = g(x)) = f(x^2) = 2(x^2) = 2x^2#

For your problem, you get (recall properties of logarithms):

#f(g(x)) = (f@g)(x) = ln(0.5(e^(3x))) = ln(0.5) + ln(e^(3x))#

#= ln(0.5) + 3x#

#= color(blue)(3x - ln2)#

And the other one (recall properties of exponents, and again, recall properties of logarithms):

#g(f(x)) = (g@f)(x) = e^(3[ln(0.5x)]) = e^(3[ln(0.5) + lnx])#

#= e^(3ln(0.5)) e^(3lnx)#

#= e^(ln(0.5^3)) e^(lnx^3)#

#= 0.5^3*x^3#

#= color(blue)(0.125x^3)#