Pre-sessionals - Maths 1

Introduction to Functions and Linear Equations

• Introduction to Functions

  • What is a function?
    • Definition: A function is a relation between a set of inputs (domain) and a set of possible outputs (range), such that each input is related to exactly one output.
    • Example: Temperature conversion function.
    • Exercise: Identify whether given relations are functions or not.
  • Domain and Range of a function
    • Definition of domain and range.
    • Example: Find the domain and range of the function \(f(x) = \sqrt{x}\).
    • Exercise: Determine the domain and range of a given function.
  • Notation and Terminology
    • Notation: \(f(x)\) represents the output of function \(f\) for input \(x\).
    • Terminology: Input, output, independent variable, dependent variable, etc.
    • Exercise: Translate word problems into function notation.

• Linear Functions

  • Definition of Linear Functions
    • A linear function is a function whose graph is a straight line.
    • Example: \(f(x) = 2x + 3\) is a linear function.
    • Exercise: Determine whether given functions are linear or not.
  • Graphing Linear Functions
    • Plotting points and connecting with a line.
    • Example: Graph the function \(f(x) = -0.5x + 2\).
    • Exercise: Graph a set of linear functions.
  • Slope and \(y\)-Intercept
    • Definition of slope and \(y\)-intercept.
    • Calculation of slope and \(y\)-intercept from an equation.
    • Example: Find the slope and \(y\)-intercept of \(f(x) = 3x - 1\).
    • Exercise: Calculate slope and \(y\)-intercept of given functions.

• Polynomial Functions of Degree 2

  • Definition of Polynomial Functions
    • A polynomial function is a function consisting of terms with non-negative integer powers.
    • Example: \(f(x) = 2x^2 - 4x + 1\) is a polynomial function.
    • Exercise: Identify polynomial functions among given expressions.
  • Quadratic Functions
    • Definition and standard form: \(f(x) = ax^2 + bx + c\).
    • Example: Identify coefficients of \(a\), \(b\), and \(c\) in \(f(x) = 5x^2 - 2x + 7\).
    • Exercise: Write quadratic functions in standard form.
  • Graphing Quadratic Functions
    • Plotting quadratic curves.
    • Finding vertex and axis of symmetry.
    • Example: Graph \(f(x) = x^2 - 4x + 3\) and find its vertex.
    • Exercise: Graph given quadratic functions and locate their vertices.

• Applications of Functions

  • Real-World Examples of Functions
    • Distance-time and temperature-time functions.
    • Example: Express the height of an object in terms of time.
    • Exercise: Identify functions in everyday scenarios.
  • Modeling with Linear and Quadratic Functions
    • Using linear functions for proportional relationships.
    • Using quadratic functions for parabolic motion.
    • Example: Model the height of a ball thrown vertically upward.
    • Exercise: Model a real-world scenario using a linear or quadratic function.
  • Simple Problems Involving Functions
    • Solving basic problems using functions.
    • Example: Find the time when a car reaches a certain distance.
    • Exercise: Solve problems involving linear and quadratic functions.

• Conclusion and Recap

  • Summarize the key concepts covered in the lecture.
  • Highlight the importance of functions in various fields.

• Q&A Session

• Exercises

• References