Question

Let f(x) = 3x + 4 and g(x) = x² + 1

Part 1: find composed functions

Part 2: give an example of a value for x where f(g(x)) is greater than g(f(x)). Give another example of a value for x where g(f(x)) is greater that f(g(x))

Make sure functions are in simplest form

Answer 1

To find the composed functions, substitute g(x) into f(x) and f(x) into g(x). Then, compare f(g(x)) and g(f(x)) by choosing different values of x.

Step-by-step explanation:

Part 1:

To find the composed functions, we substitute g(x) into f(x) and f(x) into g(x).

f(g(x)) = f(x² + 1) = 3(x² + 1) + 4 = 3x² + 3 + 4 = 3x² + 7

g(f(x)) = g(3x + 4) = (3x + 4)² + 1 = 9x² + 24x + 16 + 1 = 9x² + 24x + 17

Part 2:

To find a value of x where f(g(x)) is greater than g(f(x)), we need to compare the two expressions:

f(g(x)) = 3x² + 7

g(f(x)) = 9x² + 24x + 17

For an example where f(g(x)) is greater, we can choose x = 1:

f(g(1)) = 3(1)² + 7 = 10

g(f(1)) = 9(1)² + 24(1) + 17 = 50

For an example where g(f(x)) is greater, we can choose x = 0:

f(g(0)) = 3(0)² + 7 = 7

g(f(0)) = 9(0)² + 24(0) + 17 = 17