Question 814799
{{{f(x)=4x^2-3}}}
Is {{{g(x)=3-(1/2)x^2}}} or {{{g(x)=3-1/2x^2}}}? If it is the first one, then use parentheses to show that the {{{x^2}}} is not in the denominator: g(x)=3-(1/2)x^2. I am going to assume that the first one is correct.<br>
(f of g)(x) means f(g(x)):
(f of g)(x) = f(g(x))
(f of g)(x) = {{{f(3-(1/2)x^2)}}}
(f of g)(x) = {{{4(3-(1/2)x^2)^2-3}}}
Now we simplify. To square g(x) we can use FOIL or the {{{(a-b)^2 = a^2-2ab_b^2}}} pattern. I prefer to use the pattern:
(f of g)(x) = {{{4((3)^2-2(3)((1/2)x^2)+ ((1/2)x^2)^2)-3}}}
Simplifying...
(f of g)(x) = {{{4(9-3x^2+ (1/4)x^4)-3}}}
Multiply by 4 using the Distributive Property:
(f of g)(x) = {{{36-12x^2+ x^4-3}}}
Re-arranging the terms and adding the constant terms:
(f of g)(x) = {{{x^4-12x^2+ 33}}}
(f of g)(4) = {{{(4)^4-12(4)^2+ 33}}}
Simplifying:
(f of g)(4) = {{{256-12(16)+ 33}}}
(f of g)(4) = {{{256-192+ 33}}}
(f of g)(4) = 97<br>
(g of f(x) = g(f(x))
(g of f(x) = {{{g(4x^2-3)}}}
(g of f(x) = {{{3-(1/2)(4x^2-3)^2}}}
Using the pattern again:
(g of f(x) = {{{3-(1/2)((4x^2)^2-2(4x^2)(3)+(3)^2)}}}
Simplifying...
(g of f(x) = {{{3-(1/2)(16x^4-24x^2+9)}}}
Distributing the 1/2:
(g of f(x) = {{{3-(8x^4-12x^2+9/2)}}}
Subtracting:
(g of f(x) = {{{3-8x^4+12x^2-9/2)}}}
Simplifying...
(g of f(x) = {{{6/2-8x^4+12x^2-9/2)}}}
(g of f(x) = {{{-8x^4+12x^2-3/2)}}}
(g of f(2) = {{{-8(2)^4+12(2)^2-3/2)}}}
Simplifying...
(g of f(2) = {{{-8(16)+12(4)-3/2)}}}
(g of f(2) = {{{-128+48-3/2)}}}
(g of f(2) = {{{-80-3/2)}}}
(g of f(2) = {{{-160/2-3/2)}}}
(g of f(2) = {{{-163/2)}}}<br>
I'll leave (f of f)(1) to you.