SOLUTION: (f of g)(4) (g of f)(2) (f of f)(1) f(x)=4x^2-3 g(x)=3-1/2x^2 I've been trying to solve this equation and i have spent about two hours on it. I think my problem might just b

Algebra ->  Functions -> SOLUTION: (f of g)(4) (g of f)(2) (f of f)(1) f(x)=4x^2-3 g(x)=3-1/2x^2 I've been trying to solve this equation and i have spent about two hours on it. I think my problem might just b      Log On


   

Question 814799: (f of g)(4) (g of f)(2) (f of f)(1)
f(x)=4x^2-3
g(x)=3-1/2x^2
I've been trying to solve this equation and i have spent about two hours on it. I think my problem might just be the fraction part.
Found 2 solutions by jsmallt9, MathTherapy:
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=4x%5E2-3
Is g%28x%29=3-%281%2F2%29x%5E2 or g%28x%29=3-1%2F2x%5E2? If it is the first one, then use parentheses to show that the x%5E2 is not in the denominator: g(x)=3-(1/2)x^2. I am going to assume that the first one is correct.

(f of g)(x) means f(g(x)):
(f of g)(x) = f(g(x))
(f of g)(x) = f%283-%281%2F2%29x%5E2%29
(f of g)(x) = 4%283-%281%2F2%29x%5E2%29%5E2-3
Now we simplify. To square g(x) we can use FOIL or the %28a-b%29%5E2+=+a%5E2-2ab_b%5E2 pattern. I prefer to use the pattern:
(f of g)(x) = 4%28%283%29%5E2-2%283%29%28%281%2F2%29x%5E2%29%2B+%28%281%2F2%29x%5E2%29%5E2%29-3
Simplifying...
(f of g)(x) = 4%289-3x%5E2%2B+%281%2F4%29x%5E4%29-3
Multiply by 4 using the Distributive Property:
(f of g)(x) = 36-12x%5E2%2B+x%5E4-3
Re-arranging the terms and adding the constant terms:
(f of g)(x) = x%5E4-12x%5E2%2B+33
(f of g)(4) = %284%29%5E4-12%284%29%5E2%2B+33
Simplifying:
(f of g)(4) = 256-12%2816%29%2B+33
(f of g)(4) = 256-192%2B+33
(f of g)(4) = 97

(g of f(x) = g(f(x))
(g of f(x) = g%284x%5E2-3%29
(g of f(x) = 3-%281%2F2%29%284x%5E2-3%29%5E2
Using the pattern again:
(g of f(x) = 3-%281%2F2%29%28%284x%5E2%29%5E2-2%284x%5E2%29%283%29%2B%283%29%5E2%29
Simplifying...
(g of f(x) = 3-%281%2F2%29%2816x%5E4-24x%5E2%2B9%29
Distributing the 1/2:
(g of f(x) = 3-%288x%5E4-12x%5E2%2B9%2F2%29
Subtracting:
(g of f(x) = 3-8x%5E4%2B12x%5E2-9%2F2%29
Simplifying...
(g of f(x) = 6%2F2-8x%5E4%2B12x%5E2-9%2F2%29
(g of f(x) = -8x%5E4%2B12x%5E2-3%2F2%29
(g of f(2) = -8%282%29%5E4%2B12%282%29%5E2-3%2F2%29
Simplifying...
(g of f(2) = -8%2816%29%2B12%284%29-3%2F2%29
(g of f(2) = -128%2B48-3%2F2%29
(g of f(2) = -80-3%2F2%29
(g of f(2) = -160%2F2-3%2F2%29
(g of f(2) = -163%2F2%29

I'll leave (f of f)(1) to you.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

(f of g)(4) (g of f)(2) (f of f)(1)
f(x)=4x^2-3
g(x)=3-1/2x^2
I've been trying to solve this equation and i have spent about two hours on it. I think my problem might just be the fraction part.

g%28x%29+=+3+-+%281%2F2%29x%5E2

g%284%29+=+3+-+%281%2F2%29%284%5E2%29
g%284%29+=+3+-+%281%2F2%29%2816%29
g%284%29+=+3+-+8, or g(4) = - 5

f(g(4)) now becomes f(- 5)

Since f%28x%29+=+4x%5E2+-+3, then f%28g%284%29%29+=+f%28-+5%29+=+4%28-+5%29%5E2+-+3

+f%28g%284%29%29+=+f%28-+5%29+=+4%2825%29+-+3

f%28g%284%29%29+=+f%28-+5%29+=+100+-+3

highlight_green%28f%28g%284%29%29+=+f%28-+5%29+=+93%29

Use the same concept to solve the rest.

You can do the check!!

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