Maths 1B: Exploring Functions

Study Notes

Maths 1B: Exploring Functions

As you delve into your second semester of college mathematics, known as Maths 1B, we'll explore one of its core concepts –– functions! In this lesson, I aim to uncover their foundations with clear explanations and real-world examples. Get ready to learn how these fundamental mathematical entities can help us model situations, solve problems, and make predictions across various disciplines.

What is a Function?

A function is defined as a rule assigning each element from the first set (called the domain) to exactly one unique value from the second set (known as the range). This assignment relationship between input values and output results allows us to represent complex systems via simple formulas.

Consider the following example: If we have a function (f(x)=2x+1), when we put (x=1), it returns (y = 2 \times 1 + 1 = 3). Thus, our point ((1,3)) belongs to the graph of the function.

Common Types of Functions

Several types of functions commonly used in problem solving include:

Linear Functions: These take the general form of (f(x) = mx + b), where (m) denotes slope and (b) represents y-intercept. For instance, if we have a straight line passing through points ((1,2)) and ((2,5)), finding the equation becomes (2 = m \times 1 + b), (5 = m \times 2 + b), resulting in (m = 1) and (b = -1), thus (f(x) = x - 1).
Quadratic Functions: These exhibit parabolic curves taking the form of (f(x) = ax^2 + bx + c), defining quadratic expressions such as (x^2 + 2x + 1). Graphically, they may either open upwards like (f(x) = x^2) or downwards like (f(x) = -x^2).
Absolute Value Functions: Defined by equations like (f(x) = |ax| + b) or (f(x) = |ax + b|), which yield symmetric reflection around their vertical axis, containing discontinuities at their points of changeover.

Exploring different kinds of functions helps students understand various phenomena more profoundly.

Composition of Functions and Transformations

Function composition refers to creating new functions using existing ones. Suppose we have two functions, (f(x)) and (g(x)). Their composition, denoted by (h(x) = f(g(x))), creates a third function. By applying transformations to a given function, for instance, shifting, scaling, or reflecting, we create new versions of the original function.

For example, let's consider the linear function (f(x) = 2x). We could apply horizontal shift by adding a constant (a): (g(x) = 2x + a). Then, composing them would result in (h(x) = f(g(x)) = 2(2x + a) = 4x + 2a), showing a doubled slope and shifted horizontally by (a) units compared to the initial function.

In conclusion, mastering functions in Maths 1B will equip you with essential tools needed for modeling real-life scenarios and interpreting data accurately. Start practicing writing functions in various forms, performing transformations, and exploring their properties to develop strong skills in this area. Happy learning!

Description

Delve into the core concept of functions in your Maths 1B course with clear explanations and practical examples, enabling you to model situations, solve problems, and make predictions across disciplines. Learn about different types of functions like linear, quadratic, and absolute value functions, along with function composition and transformations.