SOLUTION: I am working with Inverse functions and need help find (f(g))(x) and (g(f))(x) given f(x)=log(2)x g(x)=2 to x power sorry if that is written confusing.. on paper the 2 after log i

Algebra ->  Algebra  -> College  -> Linear Algebra -> SOLUTION: I am working with Inverse functions and need help find (f(g))(x) and (g(f))(x) given f(x)=log(2)x g(x)=2 to x power sorry if that is written confusing.. on paper the 2 after log i      Log On


   

Question 145524: I am working with Inverse functions and need help find (f(g))(x) and (g(f))(x) given f(x)=log(2)x g(x)=2 to x power
sorry if that is written confusing.. on paper the 2 after log is lowered
Answer by jim_thompson5910(21667) About Me  (Show Source):
You can put this solution on YOUR website!
Let's evaluate f(g(x))

f%28x%29=log%282%2C%28x%29%29 Start with the first function


f%28g%28x%29%29=log%282%2C%282%5Ex%29%29 Plug in g%28x%29=2%5Ex


f%28g%28x%29%29=x%2Alog%282%2C%282%29%29 Rewrite the right side using the identity log%28b%2C%28x%5Ey%29%29=y%2Alog%28b%2C%28x%29%29


f%28g%28x%29%29=x%2A1 Take the log base 2 of 2 to get 1. Note: log%28x%2C%28x%29%29=1


f%28g%28x%29%29=x Multiply



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Let's evaluate g(f(x))

g%28x%29=2%5Ex Start with the second function


g%28f%28x%29%29=2%5E%28log%282%2C%28x%29%29%29 Plug in f%28x%29=log%282%2C%28x%29%29



Let y=log%282%2C%28x%29%29. So this means that 2%5Ey=x by use of the property log%28b%2C%28x%29%29=y ====> b%5Ey=x


g%28f%28x%29%29=2%5Ey Replace log%282%2C%28x%29%29 with "y"


g%28f%28x%29%29=x Now replace 2%5Ey with "x"




So both f(g(x)) and g(f(x)) equal x.