SOLUTION: Given the functions f(x)=1/x-3 ; g(x)=5x/x+4 Find a.f(5)+g(-3) b.(f+g)(x) c.(g∘f)(x) d.(f∘g)(-2)

Algebra ->  Rational-functions -> SOLUTION: Given the functions f(x)=1/x-3 ; g(x)=5x/x+4 Find a.f(5)+g(-3) b.(f+g)(x) c.(g∘f)(x) d.(f∘g)(-2)       Log On


   

Question 775839: Given the functions
f(x)=1/x-3 ; g(x)=5x/x+4
Find
a.f(5)+g(-3)
b.(f+g)(x)
c.(g∘f)(x)
d.(f∘g)(-2)
Answer by tanjo3(60) About Me  (Show Source):
You can put this solution on YOUR website!
I will assume the functions are as follows:
f%28x%29=1%2F%28x-3%29; g%28x%29=5x%2F%28x%2B4%29

a. f(5) + g(-3)
f%285%29=+1+%2F+%285-3%29
f%285%29=+1%2F2

g%28-3%29=+5%28-3%29+%2F+%28-3%2B4%29
g%28-3%29=+-15

f%285%29+%2B+g%28-3%29+=+1%2F2+%2B+%28-15%29
f%285%29+%2B+g%28-3%29+=+1%2F2+%2B+%28-30%29+%2F+2
f%285%29+%2B+g%28-3%29+=+-29%2F2

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b. (f + g)(x)
%28f+%2B+g%29%28x%29+=+f%28x%29+%2B+g%28x%29
%28f+%2B+g%29%28x%29+=+1%2F%28x-3%29+%2B+5x%2F%28x%2B4%29
%28f+%2B+g%29%28x%29+=+%28%28x%2B4%29%2B%285x%29%28x-3%29%29+%2F+%28%28x-3%29%28x%2B4%29%29
%28f+%2B+g%29%28x%29+=+%285x%5E2+-+14x+%2B+4%29+%2F+%28x%5E2+%2B+x+-+12%29

-------------------------------------------------------------------------------

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%28-2%29=+5%28-2%29+%2F+%28-2%2B4%29
g%28-3%29=+%28-10%29+%2F+2
g%28-3%29=+-5

(f ∘ g)(-2) = f(-5)
(f ∘ g)(-2) = 1 / (-5-3)
(f ∘ g)(-2) = -1/8