Question 775839
I will assume the functions are as follows:
{{{f(x)=1/(x-3)}}}; {{{g(x)=5x/(x+4)}}}
 
a. f(5) + g(-3)
{{{f(5)= 1 / (5-3)}}}
{{{f(5)= 1/2}}}
 
{{{g(-3)= 5(-3) / (-3+4)}}}
{{{g(-3)= -15}}}
 
{{{f(5) + g(-3) = 1/2 + (-15)}}}
{{{f(5) + g(-3) = 1/2 + (-30) / 2}}}
{{{f(5) + g(-3) = -29/2}}}
 
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b. (f + g)(x)
{{{(f + g)(x) = f(x) + g(x)}}}
{{{(f + g)(x) = 1/(x-3) + 5x/(x+4)}}}
{{{(f + g)(x) = ((x+4)+(5x)(x-3)) / ((x-3)(x+4))}}}
{{{(f + g)(x) = (5x^2 - 14x + 4) / (x^2 + x - 12)}}}
 
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c. (g ∘ f)(x)
(g ∘ f)(x) = g(f(x))
(g ∘ f)(x) = g(1/(x-3)))
(g ∘ f)(x) = [5(1/(x-3))]/[(1/(x-3))+4]
(g ∘ f)(x) = [5/(x-3)]/[(1/(x-3))+((4(x-3))/(x-3))]
(g ∘ f)(x) = [5/(x-3)]/[(4x-11)/(x-3)]
(g ∘ f)(x) = (4x-11)/5
 
Note: (g ∘ f)(x) = (4x-11)/5 for x ≠ 3
 
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d. (f ∘ g)(-2)
(f ∘ g)(-2) = f(g(-2))
 
{{{g(-2)= 5(-2) / (-2+4)}}}
{{{g(-3)= (-10) / 2}}}
{{{g(-3)= -5}}}
 
(f ∘ g)(-2) = f(-5)
(f ∘ g)(-2) = 1 / (-5-3)
(f ∘ g)(-2) = -1/8