## 1. Topic Overview

Functions are mathematical entities that assign unique outputs to specific inputs. They play a critical role in various areas of mathematics and are essential for understanding more complex concepts in higher-grade mathematics.

### Main Concept/Theme

- Understanding the definition and properties of functions.
- Analyzing different types of functions and their graphs.
- Learning how to solve problems involving functions.

### Key Learning Objectives

- Define a function and understand its notation.
- Identify and interpret different types of functions: linear, quadratic, polynomial, exponential, and logarithmic.
- Graph functions and analyze their behavior.
- Perform operations on functions, including finding the inverse of functions.

## 2. Key Terms and Definitions

**Function:**A relation between a set of inputs and a set of possible outputs where each input is related to exactly one output.**Domain:**The set of all possible input values (x-values) for the function.**Range:**The set of all possible output values (y-values) for the function.**Linear Function:**A function of the form f(x) = mx + c, where m and c are constants.**Quadratic Function:**A function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants.**Polynomial Function:**A function that is the sum of terms of the form ax^n where n is a non-negative integer.**Exponential Function:**A function of the form f(x) = a*b^x, where a and b are constants, and b > 0.**Logarithmic Function:**The inverse of the exponential function, written as f(x) = log_b(x).

## 3. Main Content Sections

### Definition and Notation of Functions

A function ( f ) from a set ( A ) (called the domain) to a set ( B ) (called the codomain) is a rule that assigns each element ( x ) in ( A ) to exactly one element ( y ) in ( B ). This is written as ( f: A \to B ).

### Types of Functions

**Linear Functions**- Form: ( f(x) = mx + c )
- Graph: A straight line with slope ( m ) and y-intercept ( c ).
**Quadratic Functions**- Form: ( f(x) = ax^2 + bx + c )
- Graph: A parabola opening upwards if ( a > 0 ) and downwards if ( a < 0 ).
**Polynomial Functions**- General form: ( f(x) = a_nx^n + a_{n-1}x^{n-1} + … + a_1x + a_0 )
- Behavior: Varies based on the degree and leading coefficient.
**Exponential Functions**- Form: ( f(x) = a \cdot b^x )
- Graph: Exhibits rapid growth for ( b > 1 ) or rapid decay for ( 0 < b < 1 ).
**Logarithmic Functions**- Form: ( f(x) = \log_b(x) )
- Graph: The inverse of the exponential function, growing slowly for large values of ( x ).

### Graphing Functions

- Use Cartesian planes to plot the points for each function.
- Identify and mark important features like intercepts, turning points, and asymptotes.
- Understand the behavior of the function at infinity and near critical points.

### Operations on Functions

**Addition:**( (f + g)(x) = f(x) + g(x) )**Subtraction:**( (f – g)(x) = f(x) – g(x) )**Multiplication:**( (fg)(x) = f(x) \cdot g(x) )**Division:**( \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}, ) where ( g(x) \neq 0 )**Composition:**( (f \circ g)(x) = f(g(x)) )

### Inverse Functions

- To find the inverse function ( f^{-1}(x) ), swap ( x ) and ( y ) in the function and solve for ( y ).

## 4. Example Problems or Case Studies

### Example 1: Linear Function

Given the function ( f(x) = 2x + 3 ):

– Identify the slope ( m = 2 ) and y-intercept ( c = 3 ).

– Graph the function by plotting the y-intercept and using the slope.

### Example 2: Quadratic Function

Given the function ( f(x) = x^2 – 4x + 3 ):

– Identify the coefficients: ( a = 1, b = -4, c = 3 ).

– Find the vertex: ( x = -\frac{b}{2a} = 2 ), then ( y = f(2) = -1 ).

– Factorize and solve for x-intercepts: ( (x – 1)(x – 3) = 0 \implies x = 1, 3 ).

## 5. Summary or Review Section

- A function relates inputs to unique outputs.
- Different types of functions have different forms and behaviors.
- Graphing functions involves plotting points and understanding the function’s characteristics.
- Operations on functions include addition, subtraction, multiplication, division, and composition.
- Inverse functions reverse the roles of inputs and outputs.

## 6. Self-Assessment Questions

- Define a function and give an example.
- What is the difference between the domain and range of a function?
- Plot the graph of the linear function ( f(x) = 3x – 2 ).
- Determine the vertex and x-intercepts of the quadratic function ( f(x) = x^2 – 6x + 5 ).
- Solve for ( y ) if ( f(x) = 2x + 1 ) and ( y = f^{-1}(x) ).

## 7. Connections to Other Topics/Subjects

- Functions are foundational for calculus, particularly limits, derivatives, and integrals.
- They are applied in various real-world contexts such as physics (motion equations), economics (cost functions), and biology (population growth models).

## Feedback Mechanism

Test your understanding by solving the self-assessment questions and checking your solutions. If you encounter difficulties, seek additional help from your teacher or reference further materials available in your textbook.