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find (f @ g)(x) and (g @ f)(x) fo the indicated functions

find (f @ g)(x) and (g @ f)(x) fo the indicated functions :
 
f(x) = 6x - 7,  g(x) = x + 7 
                                  6               
 
(f @ g)(x) = ?
 
(g @ f)(x) = ?    
 
simplify your answers
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2 Answers

Hi Keith!
 
When you see problems like (f@g)(x), read it as f(g(x)). The problem tells you what f(x) and g(x) are, therefore f(g(x)) means you're taking x of the given f(x) equation and substituting the x for g(x). 
 
(F@g)(x) means "Instead of the usual F(x), which reads "F as a function of x", you'll solve for F as a function of g(x)", or let x=g(x):
 
F(x)=6x-7 and G(x)=(x+7)/6 …so… F(g(x))=6[(x+7)/6]-7
 
Now, distribute what you have on the right side of the equation (which I'll let you do on your own), and you'll have your answer!
 
Now, for the second half of this problem, use this same method to solve for G(f(x)).
 
**Important Note: With this problem, F(g(x)) and G(f(x)) just happen to be the same, but this is not a typical case! Make sure you do the work on both parts of the problem.
 
I find color coding really helps some people when working on these types of problems, like I used in my answer. I hope this helps you, and let me know if I can help you with anything else.
 
Good luck on the rest of your assignment :)
 
 
 
Hi again Keith;
f(x)=6x-7
f(g(x))={(6)[(x+7)/6]}-7
The 6 in both the numerator and denominator cancels...
f(g(x))=(x+7)-7
f(g(x))=x
 
g(x)=(x+7)/6
g(f(x))=[(6x-7)+7]/6
g(f(x))=(6x-7+7)/6
g(f(x))=6x/6
g(f(x))=x
 
While we have identical results herein, this is not typical of f(g(x)) and g(f(x)).  Please do not always expect such.

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