Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Determine domain and range of linear, quadratic, and hyperbolic functions.
• Represent domain and range using interval and set builder notation.
• Identify restrictions on the domain of a function.

Lesson summary

This week, we delve deeper into the fascinating world of functions, building upon the foundations laid in the previous weeks. We will focus on understanding the domain and range of different types of functions. Functions are not just abstract mathematical concepts; they model real-world relationships, from calculating the cost of airtime based on usage to understanding the trajectory of a soccer ball. Understanding functions and their properties, especially domain and range, helps us make sense of these relationships and solve problems accurately. For instance, imagine trying to calculate the profits of a small spaza shop.

Lesson notes

What are Domain and Range?

Domain: The domain of a function is the set of all possible input values (x-values) for which the function is defined. Think of it as all the "allowed" values you can plug into the function.

Range: The range of a function is the set of all possible output values (y-values) that the function can produce. It's the result of plugging in all the values from the domain.

Representing Domain and Range: We can represent the domain and range in a few ways: Set Builder Notation: This uses inequalities and a set definition.

For example: {x | x ∈ ℝ, x > 2} means "the set of all x such that x is a real number and x is greater than 2." "∈ ℝ" means "is an element of the real numbers." Interval Notation: This uses brackets and parentheses to show intervals of numbers.

For example: (2, ∞) means "all numbers greater than 2, but not including 2." [2, ∞) means "all numbers greater than or equal to 2." (-∞, ∞) represents all real numbers.

Graphical Representation: Observing the graph to see where it starts and ends on both the x-axis (domain) and y-axis (range). Domain and Range for Different Function Types: Linear Functions (y = mx + c): Domain:* Generally, the domain of a linear function is all real numbers, because you can plug in any value for x.

In set builder notation: {x | x ∈ ℝ}.

In interval notation: (-∞, ∞).

Range:* Similarly, the range is usually all real numbers: {y | y ∈ ℝ} or (-∞, ∞).

Exception:* If the linear function is a horizontal line (y = c), the domain is still all real numbers, but the range is just a single value: {y | y = c}.

Example: Consider y = 2x +

1. No matter what x-value we choose, we will always get a real number for y. Quadratic Functions (y = ax² + bx + c): Domain:* The domain of a quadratic function is all real numbers: {x | x ∈ ℝ} or (-∞, ∞). You can square any real number.

Range:* The range depends on whether the parabola opens upwards (a > 0) or downwards (a 0) or maximum (if a 0).