This is what I have forFor each of the following, 1- Determine the definite equation for f[g(x)] and g[f(x)]. 2- Determine the domain and range of f[g(x)] and g[f(x)]
e) f(x) = 10^x , g(x)= log x (log x base 10)
f[g(x)] = 10 ^ ( log x) = x
g[f(x)] = log 10^x = x.
f[g(x)] = {x E R , x > 0 } because in g(x) x can not be negative.......................... g[f(x)] = { x E R, x > 0 } because in g(x) x can not be negative
f[g(x)] ={y E R, y > 0}................................................ ....................................g[f(x)] = {y E R, y > 0}
However my text book says
f[g(x)] = {x E R , x > 0 } ......................g[f(x)] = { x E R }
f[g(x)] = { y E R} ...................... ... g[f(x)] ={y E R}
Who is right? Thank you in advance.
ok I think I see the confusion.
It's the subtle difference between the domain of
f(x) vs. f(g(x)) ; and
g(x) vs. g(f(x))
you don't like that g(f(x)) has domain all of R because g(x) has domain {x:x > 0}
but you don't pass x to g here.. you pass f(x). You pass x to f, and f has domain of all of R.
the second thing you don't agree with is that the range of g(f(x)) is all R and not just {y:y>0}
let u=f(x)=10^{x}
well log(u) is negative for {u<1}. {u<1} is certainly in the range of f(x). It occurs for {x<0}
so the range of g(f(x)) is in fact all of R