Question

B. Let f:x=3x² +1 and g: x=x-5
Find
i. fog(x)
ii. go f(x)
iii. (ƒ o g)−¹(x)

Answer 1

b.

i. To find fog(x), we need to first apply g to x and then apply f to the result.

g(x) = x - 5

So,

fog(x) = f(g(x)) = f(x - 5)

= 3(x - 5)² + 1

= 3(x² - 10x + 25) + 1

= 3x² - 30x + 76

Therefore, fog(x) = 3x² - 30x + 76.

ii. To find gof(x), we need to first apply f to x and then apply g to the result.

f(x) = 3x² + 1

So,

gof(x) = g(f(x)) = g(3x² + 1)

= 3x² + 1 - 5

= 3x² - 4

Therefore, gof(x) = 3x² - 4.

iii. We want to find the inverse of the composite function f o g.

Let y = fog(x) = f(g(x)) = f(x - 5) = 3(x - 5)² + 1

To find the inverse of f o g, we need to solve for x in terms of y.

3(x - 5)² + 1 = y

3(x - 5)² = y - 1

(x - 5)² = (y - 1) / 3

x - 5 = ±√((y - 1) / 3)

x = 5 ±√((y - 1) / 3)

Therefore, the inverse of f o g is:

(ƒ o g)−¹(x) = 5 ±√((x - 1) / 3)