![Functions
A function 𝑓 is a rule that assigns to every number 𝑥 in a collection D,
a number 𝑓(𝑥). The set D is called the domain of the function. And 𝑓(𝑥
) is called the value of a function at 𝑥, or commonly known as range.
The set of ordered pair (𝑥, 𝑓(𝑥)) is called the graph of 𝑓.
• 𝑓(𝑥) = 𝑥2 + 2 is a function; it yields exactly one possible value.
• [𝑓(𝑥)]2 = 𝑥 + 2 is not a function; at specific value of x, there are
two values](https://image.slidesharecdn.com/2functionsandlimits-230922030011-a9f136c3/85/Functions-and-Limits-3-320.jpg)
Functions and Limits
- 1. Functions, Limits, and Graphs CEE 101 – Engineering Calculus 1
- 2. Definition of Terms • Domain - is the set of all values that can be plugged into a function and have the function exist and have a real number for a value. • Range – is the set of all possible values that a function can take. • Variable - It is a quantity which, during any set of mathematical operations, does not retain the same value but is capable of assuming different values. • Constant - It is a quantity which, during any set of mathematical operations, retains the same value. • Polynomial - It can have constants, variables, and exponents, but never division by a variable • Conjugate - It is formed by changing the sign between two terms.
- 3. Functions A function 𝑓 is a rule that assigns to every number 𝑥 in a collection D, a number 𝑓(𝑥). The set D is called the domain of the function. And 𝑓(𝑥 ) is called the value of a function at 𝑥, or commonly known as range. The set of ordered pair (𝑥, 𝑓(𝑥)) is called the graph of 𝑓. • 𝑓(𝑥) = 𝑥2 + 2 is a function; it yields exactly one possible value. • [𝑓(𝑥)]2 = 𝑥 + 2 is not a function; at specific value of x, there are two values
- 4. Limits It is a fundamental concept used to describe the behavior of a function as its input approaches a certain value. The limit of a function 𝑓(𝑥) as 𝑥 approaches 𝑎 is 𝐿, can be written as lim 𝑥→𝑎 𝑓 𝑥 = 𝐿 In other words, the value of the function 𝑓(𝑥) gets closer and closer to 𝐿 as 𝑥 gets closer and closer to 𝑎, without being exactly equal to 𝑎.
- 5. Limits Example 1: lim 𝑥→1 2𝑥 + 5 = ?
- 9. Evaluate: lim 𝑥→2 4𝑥2 − 1 lim 𝑥→−3 𝑥2 − 9 𝑥 + 3
Editor's Notes
- #5: Why do we have to study limits? - Because, essentially, that’s what calculus is: the study of limits. Limits are used to examine function behavior around a specific point, and without it, it would be very difficult to talk about rate of change. Therefore, calculus is impossible to do without knowing what to do with limits.