# Let f(x)= - X^2 - 4 x - 1 and g(x) = - 3 X^2 + 5 x + 4 . Find g(f(0)) .

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f(x) = -x^2 - 4x -1

g(x) = -3x^2 + 5x + 4

find g(f(0)

First we need to determine g(f(x)

g(f(x) = g ( -x^2 - 4x -1)

= -3(x^2-4x-1)^2 + 5(-x^2 - 4x -1) + 4

Let us factor (-x^2 - 4x -1) :

= ( -x^2 - 4x -1) [ -3(x^2 - 4x -1) + 5] + 4

= ( -x^2 - 4x -1) ( -3x^2 +12x + 3 + 5) + 4

= ( -x^2 - 4x -1) ( -3x^2 + 12x + 8) + 4

Now we will substitute with x = 0:

==> g(f(0) = ( -0 -0 -1) ( -0+0 + 8) + 4

= -1* 8 + 4

= -8 + 4 = -4

==>** g(f(0)) = -4**

We are given the functions f(x)= - x^2 - 4 x - 1 and g(x) = - 3 x^2 + 5 x + 4 and we have to find g(f(0)).

For this we first need to find f(0).

Now f(x)= - x^2 - 4 x - 1 , substituting x = 0 we get f(0) = -1.

Now substitute x= -1 in g(x)

=> g(x) = - 3 x^2 + 5 x + 4

=> g(-1) = - 3 *(-1)^2 + 5(-1) + 4

=> g(-1) = -3 -5 + 4

=> g(-1) = -4

**Therefore we get that g(f(0)) = -4**

5(-1) f(x) = -x^2-4x-1.

g(x) = -3x^2+5x+4.

To find g(f(0)).

We first find the function g(f(x)):

g(x) = -3x^2+5x+4.

Therefore g(f(x)) = -3f(x)+5f(x)+4.

Now we put f(x) = -x^2-4x-1 in -3f(x)+5f(x)+4.

g(f(x)) = -3(- X^2 - 4 x - 1)^2+5(- X^2 - 4 x - 1)+4.

Also f(0) = (-0^2-4*0-1) = -1.

Therefore g(f(0)) = -3(f(0))^2 +5f(0) +4.

g(f(0)) = -3 (-1)^2 +5(-1)+4 = -3-5+4 = -4.

Therefore g(f(0)) = -4.