SOLUTION: 8. a) Given the functions f(x) = x + 2 and g(x) = 3^x, determine an equation for (f ∘ g)(x) and (g ∘ f)(x). b) Determine f(g(3)) and g(f(3)). c) Determine all values of x f

(f ∘ g)(x) means to replace the letter x in the right side of f(x) with the
right side of g(x).  So we start with f

f(x) = x + 2

and replace the x by the right side of g(x) which is 3x, so we
have:

(f ∘ g)(x) = 3x + 2

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(g ∘ f)(x) means to replace the letter x in the right side of g(x) with the
right side of f(x).  So we start with 

g(x) = 3x

and replace the x by the right side of f(x) which is x + 2, so we have:

(g ∘ f)(x) = 3x + 2 

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There is no difference between f(g(3)) and  (f ∘ g)(3).  They are one in the
same.  So f(g(3)) means to replace x by 3 in the equation for
(f ∘ g)(x).  So we take the equation for (f ∘ g)(x) which is

(f ∘ g)(x) = 3x + 2

and replace x by 3

(f ∘ g)(3) = 33 + 2

and simplify:

(f ∘ g)(3) = 3∙3∙3 + 2

(f ∘ g)(3) = 27 + 2

(f ∘ g)(3) = 29

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There is no difference between g(f(x)) and  (g ∘ f)(x).  They are one in the
same. 

So g(f(3)) means to replace x by 3 in the equation for 
(g ∘ f)(x).  So we take the equation for (g ∘ f)(x) which is

(g ∘ f)(x) = 3x + 2

and replace x by 3

(g ∘ f)(3) = 33 + 2

and simplify:

(g ∘ f)(3) = 35

(g ∘ f)(3) = 3∙3∙3∙3∙3

(g ∘ f)(3) = 243

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That's the same as:

   Determine all values of x for which (f ∘ g)(x) = (g ∘ f)(x) 

So we set the right side of (f ∘ g)(x) equal to the right side of 
(g ∘ f)(x):

(f ∘ g)(x) = (g ∘ f)(x) 

3x + 2 = 3x + 2

and simplify:

3x + 2 = 3x∙32

3x + 2 = 3x∙9 

Swap sides:

3x∙9 = 3x + 2 

9∙3x = 3x + 2

9∙3x - 3x = 2 

8∙3x = 2

3x = 2/8

3x = 1/4

Take logs of both sides:

log(3x) = log(1/4)

x∙log(3) = log(1/4)

x = log(1/4)/log(3)

x = -0.6020599913/0.4771212547

x = -1.261859507    <--the only value of x for which f(g(x)) = g(f(x)). 

Edwin