For all values of x

(2)

f (x) = (x + 2) 2 and g (x) = 3 (x - 1)

a) Find gf (x), giving your answer in the form a (x2 + bx + c)

where a, b and care integers.

(x² +

x +

b) Find g ' (12)

(2)

A.) gf (x) = 3x^2 + 12x + 9

B.) g' (x) = 2

Step-by-step explanation:

A.) The two given functions are:

f (x) = (x + 2) ^2 and g (x) = 3 (x - 1)

Open the bracket of the two functions

f (x) = (x + 2) ^2

f (x) = x^2 + 2x + 2x + 4

f (x) = x^2 + 4x + 4

and

g (x) = 3 (x - 1)

g (x) = 3x - 3

To find gf (x), substitute f (x) for x in g (x)

gf (x) = 3 (x^2 + 4x + 4) - 3

gf (x) = 3x^2 + 12x + 12 - 3

gf (x) = 3x^2 + 12x + 9

Where

a = 3, b = 12, c = 9

B.) To find g ' (12), you must first find the inverse function of g (x) that is g' (x)

To find g' (x), let g (x) be equal to y. Then, interchange y and x for each other and make y the subject of formula

Y = 3x + 3

X = 3y + 3

Make y the subject of formula

3y = x - 3

Y = x/3 - 3/3

Y = x/3 - 1

Therefore, g' (x) = x/3 - 1

For g' (12), substitute 12 for x in g' (x)

g' (x) = 12/4 - 1

g' (x) = 3 - 1

g' (x) = 2.