Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Define a function and distinguish it from a relation.
• Represent functions as equations, tables, diagrams, and graphs.
• Identify the domain and range of a function.
• Use the vertical line test to identify functions.
• Use function notation, like f(x), to evaluate functions.

Lesson summary

Functions are a fundamental concept in mathematics, forming the basis for understanding relationships between variables and modeling real-world phenomena.

Think about the price of airtime: the more airtime you buy, the more it costs. This relationship can be modeled as a function. Functions are also crucial for understanding everything from the spread of disease to the trajectory of a soccer ball. In the South African context, functions can be used to model things like electricity consumption based on household size, the growth of small businesses based on investment, or even the relationship between rainfall and crop yield.

Lesson notes

What is a Function? A function is a special type of relation where each input (x-value) has exactly one output (y-value).

Imagine a vending machine: when you press a specific button (input), you expect to get only one specific item (output). If pressing the button for Coke sometimes gives you Coke and sometimes gives you Fanta, it's not a function!

Relation: Any set of ordered pairs (x, y).

Function: A relation where each x-value corresponds to only one y-value.

Domain: The set of all possible input values (x-values).

Range: The set of all possible output values (y-values). Representing Functions Functions can be represented in several ways: Equations: An equation that defines the relationship between x and y.

Example: `y = 2x + 1` Tables: A table listing x-values and their corresponding y-values. | x | y | |---|---| | 0 | 1 | | 1 | 3 | | 2 | 5 | Mapping Diagrams: Arrows connecting x-values in one set to their corresponding y-values in another set.

Graphs: A visual representation of the function on the Cartesian plane. Function Notation We use function notation to write functions concisely. Instead of writing `y = 2x + 1`, we can write `f(x) = 2x + 1`. This reads as "f of x equals 2x plus 1." `f(x)` represents the output value of the function f for a given input x. To evaluate a function, substitute the given value for x into the function's equation. The Vertical Line Test A quick way to determine if a graph represents a function is to use the vertical line test. If any vertical line drawn on the graph intersects the graph at more than one point, then the graph does not represent a function. This is because a single x-value would have multiple corresponding y-values, violating the definition of a function.

Worked example

Example 1: Is the following relation a function? {(1, 2), (2, 4), (3, 6), (1, 5)}

Solution: No, this is not a function. The x-value 1 is paired with two different y-values (2 and 5).

Example 2: Given the function f(x) = 3x - 2, find f(4).

Solution: To find f(4), substitute x = 4 into the function:

f(4) = 3(4) - 2

f(4) = 12 - 2

f(4) = 10

Example 3: Determine the domain and range of the function represented by the following table:

| x | y |

|----|----|

| -2 | 0 |

| 0 | 2 |

| 2 | 4 |

| 4 | 6 |

Solution:

Domain: {-2, 0, 2, 4} (The set of all x-values)

Range: {0, 2, 4, 6} (The set of all y-values)