# Grade 10 Mathematics: Functions Study Notes

## 1. Topic Overview

In this section, we will explore the concept of functions, a fundamental topic in mathematics that describes the relationship between two variables. Functions are used to model real-world situations and solving problems.

### Key Learning Objectives

- Understand the definition of a function
- Differentiate between different types of functions
- Learn how to represent functions using equations, tables, and graphs
- Analyze the properties of functions such as domain and range

## 2. Key Terms and Definitions

**Function**: A relationship between two sets where each element in the first set (called the domain) is related to exactly one element in the second set (called the range).**Domain**: The set of all possible input values for a function.**Range**: The set of all possible output values of a function.**Independent Variable**: The variable that represents the input of a function (usually denoted as x).**Dependent Variable**: The variable that represents the output of a function (usually denoted as y or f(x)).**Linear Function**: A function that forms a straight line when graphed. Its general form is f(x) = mx + b.**Quadratic Function**: A function that forms a parabola when graphed. Its general form is f(x) = ax^2 + bx + c.

## 3. Main Content Sections

### 3.1 Understanding Functions

A function assigns each input exactly one output. For example, if f(x) = 2x + 3, then the function f will take any value of x, multiply it by 2, and then add 3 to find the output.

### 3.2 Representing Functions

**Equations**: Functions can be expressed algebraically by an equation, such as f(x) = 2x + 3.**Tables**: Functions can also be represented using a table of values.

| x | f(x) |

|—|——|

| 1 | 5 |

| 2 | 7 |

| 3 | 9 |

**Graphs**: Lastly, functions can be represented using graphs, showing the relationship between the dependent and independent variables.

### 3.3 Types of Functions

**Linear Functions**: These have the form f(x) = mx + b. The graph is a straight line with slope m and y-intercept b.**Quadratic Functions**: These have the form f(x) = ax^2 + bx + c. The graph is a parabola that opens upwards if a > 0 and downwards if a < 0.**Other Polynomials**: Functions can be higher-degree polynomials like cubic functions (f(x) = ax^3 + bx^2 + cx + d).

### 3.4 Properties of Functions

**Domain and Range**: The domain of f(x) = 2x + 3 is all real numbers (−∞, ∞), as x can be any number. The range is also all real numbers because the function can produce any y-value.**Intercepts**: The x-intercept is where the graph crosses the x-axis (f(x) = 0), and the y-intercept is where it crosses the y-axis (x=0).

## 4. Example

### Example 1: Linear Function

Given f(x) = 3x – 4:

1. Find f(2).

2. Determine the x-intercept.

3. Graph the function.

#### Solution:

- [ f(2) = 3(2) – 4 = 6 – 4 = 2 ]
- Set f(x) to zero: [ 0 = 3x – 4 \Rightarrow x = \frac{4}{3} ]
- The y-intercept is -4 (when x=0).

### Example 2: Quadratic Function

Given g(x) = x^2 – 6x + 8:

1. Find g(3).

2. Determine the vertex.

#### Solution:

- [ g(3) = 3^2 – 6(3) + 8 = 9 – 18 + 8 = -1 ]
- Vertex form: [ g(x) = (x-3)^2 – 1 ]

## 5. Summary

In this topic, we learned that functions are relationships between inputs and outputs. We explored how to represent functions through equations, tables, and graphs. Key types include linear and quadratic functions. We also delved into understanding the properties of functions such as domain and range.

## 6. Self-Assessment Questions

- Define a function in your own words.
- Given the function h(x) = -x^2 + 5x + 6, find h(-1).
- What is the domain of the function k(x) = 1/(x-2)?
- Sketch the graph of the function f(x) = 2x + 1.
- Determine the x-intercepts of the function j(x) = x^2 – 9.

## 7. Connections to Other Topics/Subjects

Functions are a cornerstone in mathematics, building a foundation for more advanced topics such as calculus. In science, functions model relationships between variables like speed vs. time. In economics, they represent relationships such as cost vs. production level. Understanding functions is crucial for problem-solving in various real-life contexts.

### Feedback Mechanism

After studying, try to solve example problems and self-assessment questions independently. Discuss challenging concepts with peers or teachers and seek further resources if necessary. Testing your understanding through practical applications helps solidify your knowledge.