Let #f(x)= 1-x^3# and #g(x)= 1/x#, how do you find each of the compositions?

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Answer 1 · 2015-07-07T18:49:18

Think of the #x# as just a place holder for something to get:

#(f*g)(x) = f(g(x)) = 1-(g(x))^3 = 1-(1/x)^3 = 1-1/x^3#

#(g*f)(x) = g(f(x)) = 1/f(x) = 1/(1-x^3)#

An equation like #f(x) = 1 - x^3# basically describes what you do to any thing to turn it into #f# of that thing.

You can replace #x# with any number or expression and the equation remains true.

So to find #(f*g)# just substitute the expression for #g(x)# into the equation for #f(x)# to get:

#(f*g)(x) = f(g(x)) = 1-(g(x))^3 = 1-(1/x)^3 = 1-1/x^3#

Similarly,

#(g*f)(x) = g(f(x)) = 1/f(x) = 1/(1-x^3)#