Let f(x) = x² + x + 2 and g(x) = 2x² + 5. Find f(g(x)). Show each step of your work.

Solution:

Given: Functions are f(x) = x² + x + 2 and g(x) = 2x² + 5.

f(g(x) is a composite function. The input values given to f(x) are the output values of g(x).

To find f(g(x)) replace the value of x in f(x) by g(x) = 2x² + 5 as shown below.

f(g(x)) = (2x² + 5)² + 2x² + 5 + 2

f(g(x)) = 4x⁴ + 25 + 20x² + 2x² + 5 + 2

Grouping the like terms and adding them, we get

f(g(x)) = 4x⁴ + 20x² + 2x² + 25+ 5 + 2

f(g(x)) = 4x⁴ + 22x² + 32

Let f(x) = x² + x + 2 and g(x) = 2x² + 5. Find f(g(x)). Show each step of your work.

Summary:

From the above workings: Find f(g(x)) = 4x⁴ + 22x² + 32.