Given that f(x) =x+4 and g(x)=2x+1 find

a. (f•g)(x)

b. (g•f)(x)

c. (f•g)(5)

d. (g•f)(5)

e. Is function composition always commutative?

Answer: a. (f • g)(x) This means we're composing the functions, applying g(x) first, then f(x): 1. Start with g(x): g(x) = 2x + 12. Substitute g(x) into f(x): f(g(x)) = (2x + 1) + 43. Simplify: (f • g)(x) = 2x + 5 b. (g • f)(x) This means we're applying f(x) first, then g(x): 1. Start with f(x): f(x) = x + 42. Substitute f(x) into g(x): g(f(x)) = 2(x + 4) + 13. Simplify: (g • f)(x) = 2x + 9 c. (f • g)(5) We already found (f • g)(x) = 2x + 5. Now, substitute x = 5: (f • g)(5) = 2(5) + 5 = 15 d. (g • f)(5) We already found (g • f)(x) = 2x + 9. Now, substitute x = 5: (g • f)(5) = 2(5) + 9 = 19 e. Is function composition always commutative? No, function composition is not always commutative. We saw in parts (a) and (b) that (f • g)(x) ≠ (g • f)(x). The order in which you compose functions matters. In general, function composition is commutative only for very specific cases, such as when the functions are inverses of each other.

Answered by ronalynlotecjavillo | 2024-09-06