a.) g(x^2 )

b.) g(3x^2-4)

c.) g(x-h)

d.) g(x)-g(h)

e.) (g(x+h)-g(x))/h, h ≠0

2.) G(x)=√(4-x)

a.) G(4-x)

b.) G(4-x^2 )

c.) G(4-x^4 )

d.) G(4x-x^2 )

3.) G(-x^4-4x^2)

Evaluate the function at the indicated values.

1.) g(x)=3x^2-4

a.) g(x^2 )

b.) g(3x^2-4)

c.) g(x-h)

d.) g(x)-g(h)

e.) (g(x+h)-g(x))/h, h ≠0

2.) G(x)=√(4-x)

a.) G(4-x)

b.) G(4-x^2 )

c.) G(4-x^4 )

d.) G(4x-x^2 )

3.) G(-x^4-4x^2)

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Marked as Best Answer

1)

a. 3(x^{2})^{2} - 4 = 3x^{4} - 4

b. 3(3x^{2} - 4)^{2} - 4 = 3(9x^{4} - 24x^{2} + 16) - 4 = 27x^{4} - 72x^{2} + 48 - 4 = 27x^{4} - 72x^{2} + 44

c. 3(x-h)^{2} - 4 = 3(x^{2} - 2xh + h^{2}) - 4 = 3x^{2} - 6xh + 3h^{2} - 4

d. (3x^{2} - 4) - (3h^{2} - 4) = 3x^{2} - 3h^{2} -4 + 4 = 3x^{2} - 3h^{2} = 3(x^{2}-h^{2}) = 3(x+h)(x-h)

e. (g(x+h) - g(x)) / h = (3(x+h)^{2} - 4 - (3x^{2}-4)) / h = (3(x^{2} +2xh + h^{2}) - 4 -3x^{2} + 4)) / h

(3x^{2} + 6xh + 3h^{2} -3x^{2}) / h = (6xh + 3h^{2})/h = 6x + 3h

The limit as h->0 of 6x + 3h is 6x, the first derivative of g(x) = 3x^{2} - 4.

The other problem is similar.

1)a) g(x^{2}) = 3(x^{2})^{2} - 4 = 3x^{4} - 4

b) g(3x^{2}-4) = 3(3x^{2} - 4)^{2} - 4 = 3(9x^{4} - 24 x^{2} + 16) - 4= 27x^{4} - 72x + 48- 4= 27x^{4} - 72x+44

C)g(x-h)= 3(x-h)^{2} - 4 = 3(x^{2} - 2hx + h^{2}) -4 = 3x^{2} -6hx + 3h^{2} -4

d)g(x)-g(h)= 3x^{2}-4-3h^{2}+4=3x^{2} - 3h^{2}

e)(g(x+h)-g(x))/h=((3(x+h)^{2}-4-3x^{2}+4)/h=(6xh+3h^{2})/h=6x+3h

2)4-x should be greater or equal to 0 , then x is less or equal to 4

a) G(4-x)= √(4-4+x)=√(x) where x ≥ 0

b) G(4-x^{2})=√(4-(4-x^{2})=√(x^{2})= x if and only if x ≥ 0, -x when x ≤ 0 as the square root should be always positive

c) G(4-x^{4})= √ (4-4+x^{4})=x^{2}

d) G(4x-x^{2})= √(4-4x+x^{2})=√(x-2)^{2}=x-2 where x ≥ 2 and -x+2 when x ≤ 2

e) G ( -x^{4}-4x^{2})= √( 4+x^4+4x^{2})= √(x^{2} + 2)^{2} = x^{2} + 2`

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