Functions and Limits

Introduction:

Functions are like tools we use to describe things in the world with math. They help us understand how different things are related to each other. This is really helpful when we’re studying calculus.

Concept of Function:

Leibniz, a German mathematician, first came up with the idea of a function. A function helps us understand how two things are related to each other using math. For example, the area of a square can be calculated using the formula: A = x². This means that the area “A” depends on the length of one of its sides “x”. We call this relationship a function of x, which means that when we know the value of “x”, we can use the function to find the corresponding value of “A”.

Likewise, the volume of a sphere can be calculated using a formula: V = 4/3 πr³. This means that the volume “V” depends on the radius of the sphere “r”.

A function is like a rule that connects two groups of things. Each thing in the first group is connected to only one thing in the second group. For example, a square with a certain length on each side will always have one specific area, and a sphere with a certain radius will always have one specific volume.

Definitions:

1. Function: A function is a set of rules that assigns each input value (x) to exactly one output value (y). It’s written as “f(x) = y”, where “x” is the input value, “y” is the output value, and “f” is the function.

Example: The function f(x) = 2x assigns each input value “x” to an output value “y” that is twice the input value. For example, if x = 3, then f(x) = 2(3) = 6.

2. Domain: The domain of a function is the set of all possible input values (x) that can be used with the function. It’s written as “x ∈ D”, where “D” is the domain.

Example: The domain of the function f(x) = √x is all non-negative real numbers, since you can only take the square root of non-negative numbers. So the domain is written as “x ∈ [0,∞)”.

3. Range: The range of a function is the set of all possible output values (y) that the function can produce for a given set of input values. It’s written as “y ∈ R”, where “R” is the range.

Example: The function f(x) = x² has a range of all non-negative real numbers since the square of any real number is non-negative. So the range is written as “y ∈ [0,∞)”.

Notation and Value of a Function

If a variable y changes based on a variable x in a way that for every value of x, there is only one value of y, then we say that “y is a function of x”.

A Swiss mathematician named Euler came up with a way to write this as y = f(x), which means “y is equal to f of x”. When we write functions, we usually use letters like f, g, h, F, or G.

A function can be thought of as a computing machine, represented by the letter “f”, that takes an input value “x”, processes it in a specific way, and produces a single output value “f(x)”. This output value “f(x)” is also known as the “value of f at x” or the “image of x under f”. We can represent this output value using a different letter, such as “y”, and write it as “y = f(x)”.

The variable “x” is known as the independent variable of the function “f”, while the variable “y” is known as the dependent variable of “f”. In the context of real numbers, we refer to “f” as a real-valued function of real numbers. This means that both the input “x” and the output “f(x)” are real numbers.

The variable “x” is known as the independent variable of the function “f”, while the variable “y” is called the dependent variable of “f”. In simpler terms, “x” is the input variable, and “y” is the output variable of the function “f”.

We will only focus on real numbers for both the input and output variables of the function “f”. Therefore, we call the function “f” a real-valued function of real numbers.

Solved example 1:
Find f(-2)
[latex] $$f(x) = {x^3 – 2x^2 + 4x~ – 1}.$$
Solution:
$$f(-2) = {(- 2)^3 – 2 (-2 )^2 + 4 (-2) – 1 = – 8 – 8 – 8 – 1 = -2 5}.$$

Solved example 2:
Find f(1/x), x ≠ 0
[latex] $$f(x) = {x^3 – 2x^2 + 4x~ – 1}.$$
Solution:
$$f(1/x) = {(1/x)^3 – 2(1/x)^2 + 4 (1/x) – 1 = {1\overx^3}-{2\overx^2}+{4\overx}- 1}.$$