# Composite Functions

• Dec 15th 2013, 12:29 PM
sakonpure6
Composite Functions
Quote:

For 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)
This is what I have for
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.
• Dec 15th 2013, 01:05 PM
romsek
Re: Composite Functions
Quote:

Originally Posted by sakonpure6
This is what I have for
e-1)f[g(x)] = 10 ^ ( log x) = x ..... e-2)g[f(x)] = log 10^x = x.

e-1) Domain {x E R , x > 0 } because in g(x) x can not be negative e-2) Domain { x E R, x > 0 } because in g(x) x can not be negative
e-1) Range { y E R, y > 0} e-2) Range {y E R, y > 0}

However my text book says
e-1) Domain {x E R , x > 0 } e-2) Domain { x E R }
e-1) Range { y E R} e-2) Range {y E R}

Who is right? Thank you in advance.

I'm sorry this is all very confusing the way you have it written. Could you repost this along the lines of

f(g(x)): Domain = {}, Range = {}

g(f(x)): Domain = {}, Range = {}

where you would fill in the braces.
• Dec 15th 2013, 01:23 PM
sakonpure6
Re: Composite Functions
is this better (I edited post)
• Dec 15th 2013, 01:53 PM
romsek
Re: Composite Functions
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)=10x

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