Question

Given the following two functions:

f(x) = x² + 2x
g(x) = x + 1
Evaluate each of the following six composite function operations and place your answers in the box to the right:

a) f(g(x))
b) g(f(x))
c) f(f(x))
d) g(g(x))
e) f(g(8))
f) g(f(8))

Answer 1

To evaluate the given composite functions, substitute the expressions for the functions into each other and simplify the resulting expressions.

Step-by-step explanation:

a) To evaluate f(g(x)), we substitute the expression for g(x) into f(x):

f(g(x)) = f(x + 1) = (x + 1)² + 2(x + 1) = x² + 2x + 1 + 2x + 2 = x² + 4x + 3

b) To evaluate g(f(x)), we substitute the expression for f(x) into g(x):

g(f(x)) = g(x² + 2x) = x² + 2x + 1

c) To evaluate f(f(x)), we substitute the expression for f(x) into f(x):

f(f(x)) = f(x² + 2x) = (x² + 2x)² + 2(x² + 2x) = x⁴ + 4x³ + 4x² + 4x² + 8x + 4 = x⁴ + 4x³ + 8x² + 8x + 4

d) To evaluate g(g(x)), we substitute the expression for g(x) into g(x):

g(g(x)) = g(x + 1) = (x + 1) + 1 = x + 2

e) To evaluate f(g(8)), we substitute 8 into the expression for g(x), and then substitute the result in to the expression for f(x):

f(g(8)) = f(8 + 1) = f(9) = 9² + 2(9) = 81 + 18 = 99

f) To evaluate g(f(8)), we substitute 8 into the expression for f(x), and then substitute the result into the expression for g(x):

g(f(8)) = g(8² + 2(8)) = g(64 + 16) = g(80) = 80 + 1 = 81