Evaluate the function at the indicated values.

Question

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)

Answer 1

1)a) g(x²) = 3(x²)² - 4 = 3x⁴ - 4

b) g(3x²-4) = 3(3x² - 4)² - 4 = 3(9x⁴ - 24 x² + 16) - 4= 27x⁴ - 72x + 48- 4= 27x⁴ - 72x+44

C)g(x-h)= 3(x-h)² - 4 = 3(x² - 2hx + h²) -4 = 3x² -6hx + 3h² -4

d)g(x)-g(h)= 3x²-4-3h²+4=3x² - 3h²

e)(g(x+h)-g(x))/h=((3(x+h)²-4-3x²+4)/h=(6xh+3h²)/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-x²)=√(4-(4-x²)=√(x²)= x if and only if x ≥ 0, -x when x ≤ 0 as the square root should be always positive

 

 c) G(4-x⁴)= √ (4-4+x⁴)=x²

d) G(4x-x²)= √(4-4x+x²)=√(x-2)²=x-2 where x ≥ 2 and -x+2 when x ≤ 2

e) G ( -x⁴-4x²)= √( 4+x^4+4x²)= √(x² + 2)² = x² + 2`

Answer 2

1)

a.  3(x²)² - 4  = 3x⁴ - 4

b.  3(3x² - 4)² - 4 = 3(9x⁴ - 24x² + 16) - 4 = 27x⁴ - 72x² + 48 - 4 = 27x⁴ - 72x² + 44

c.  3(x-h)² - 4 = 3(x² - 2xh + h²) - 4 = 3x² - 6xh + 3h² - 4

d.  (3x² - 4) - (3h² - 4) = 3x² - 3h² -4 + 4 = 3x² - 3h² = 3(x²-h²) = 3(x+h)(x-h)

e.  (g(x+h) - g(x)) / h = (3(x+h)² - 4 - (3x²-4))  / h  = (3(x² +2xh + h²) - 4 -3x² + 4)) / h

 

(3x² + 6xh + 3h² -3x²) / h  = (6xh + 3h²)/h  = 6x + 3h

 

The limit as h->0 of 6x + 3h is 6x, the first derivative of g(x) = 3x² - 4.

 

 

The other problem is similar.