Unit 16 – Day 1 Name: Date: Integrated Analysis: NOTES

Functions and Limits – Composition of Functions Function operations refers to the addition, subtraction, multiplication, and division of functions. You can just think of a function as a special type of equation. Function Notation: Functions are often written using “ 𝑓(𝑥) ” (which means “function of 𝑥”). It is

read as “𝑓 of 𝑥”. It is just a more useful way of writing 𝑦. We do not always have to use 𝑓(𝑥) – we may use any letter replacing the 𝑓 or 𝑥. 𝑔(𝑥) = 4𝑥 + 6 ℎ(𝑥) = 5𝑥2 − 2𝑥 + 4 𝑑(𝑡) = 6𝑡 The variable in the equation must match the variable inside of the parenthesis.

Function Operations

Operation Mathematical notation How to think of it

Addition (𝑓 + 𝑔)𝑥 𝑓(𝑥) + 𝑔(𝑥)

Subtraction (𝑓 − 𝑔)𝑥 𝑓(𝑥) − 𝑔(𝑥)

Multiplication (𝑓 ∙ 𝑔)𝑥 𝑓(𝑥) ∙ 𝑔(𝑥)

Division (𝑓𝑔) 𝑥

𝑓(𝑥)𝑔(𝑥)

Given 𝑓(𝑥) = 2𝑥 − 4 and 𝑔(𝑥) = 𝑥2 − 6𝑥 + 8 find the following: Ex 1) (𝑓 + 𝑔)𝑥 Ex 2) (𝑓 − 𝑔)𝑥 Ex 3) (𝑓 ∙ 𝑔)𝑥 Ex 4) (𝑓

𝑔) 𝑥

Unit 16 – Day 1 Composition of Functions: A way of combining two function to create a new one. Function Composition Notation

Notation #1 Notation #2 What it says to do… (𝑓 ∘ 𝑔)(𝑥) 𝑓(𝑔(𝑥)) Take 𝑔(𝑥) and plug it into 𝑓(𝑥)

(𝑔 ∘ 𝑓)(𝑥) 𝑔(𝑓(𝑥)) Take 𝑓(𝑥) and plug it into 𝑔(𝑥)

Given 𝑓(𝑥) = 𝑥 − 2 and 𝑔(𝑥) = 3𝑥 + 4, find the following: Ex 5) 𝑓(𝑔(𝑥)) Ex 6) 𝑔(𝑓(𝑥)) Ex 7) 𝑓(𝑓(𝑥)) Ex 8) 𝑔(𝑔(𝑥))