Question

How do you find the compositions given `f(x) =x^2 + 2x + 1` and `g(x) = x - 2`?

Answer 1

There isn't just one composition, but I can show you a couple different types.

Explanation:

Example: Find the value of ƒ(g(-3))

ƒ(g(-3) means that what is the value of the function ƒ when g has an x value of -3.

g(-3) = -3 - 2
g(-3) = -5
g(-3) = ƒ(-5)
ƒ(-5) = `(-5)^2` + 2(-5) + 1
ƒ(-5) = 25 - 10 + 1
ƒ(-5) = 16

The expression ƒ(g(-3)) has a value of 16.

One more example:

Find h(x) if h(x) = ƒ(g(x))

h(x) = ƒ(g(x)

You can plug function g(x) into ƒ(x):

g(x) = x - 2
ƒ(g(x) = `(x - 2)^2` + 2(x - 2) + 1
h(x) = `x^2` - 4x + 4 + 2x - 4 + 1
h(x) = `x^2` - 2x + 1
h(x) = `(x - 1)^2`

Hopefully you understand now. :)