Question

For the functions f(x)=4x−3 and g(x)=3x2+4x, find (f∘g)(x) and (g∘f)(x).

Answer 1

(g∘f)(x)=48x2+48x+10

(g∘f)(x)=12x^2-6

Explanation:

To find (f∘g)(x), use the definition of (f∘g)(x),

(f∘g)(x)=f(g(x))

Substituting 3x2−2 for g(x) gives

(f∘g)(x)=f(3x2−2)

Find f(3x2−2), where f(x)=4x+2, and simplify to get

(f∘g)(x)(f∘g)(x)(f∘g)(x)=4(3x2−2)+2=12x2−8+2=12x2−6

To find (g∘f)(x), use the definition of (g∘f)(x),

(g∘f)(x)=g(f(x))

Substituting 4x+2 for f(x) gives

(g∘f)(x)=g(4x+2)

Find g(4x+2), where g(x)=3x2−2, and simplify to get

(g∘f)(x)=3(4x+2)^2−2

(g∘f)(x)=48x2+48x+12−2

(g∘f)(x)=48x2+48x+10

Answer 2

(16x + 21) and (16x - 6)

Explanation:

f(g(x)) = f(6 + 4x)

Applying the f(x) function on (6 + 4x) gives

4(6 + 4x) - 3

Which equals 16x + 24 - 3

= 16x + 21

g(f(x)) = g(4x - 3)

Applying the g(x) function on (4x - 3) gives

6 + 4(4x - 3)

Which equals 6 + 16x - 12

= 16x - 6