Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Define a function, domain, and range.
• Represent functions in different ways.
• Determine the domain and range of basic functions.
• Use and evaluate function notation.
• Sketch basic linear and quadratic graphs.

Lesson summary

Functions are fundamental building blocks in mathematics and are crucial for understanding relationships between quantities. They allow us to model and analyze real-world situations, from predicting stock market trends to understanding the trajectory of a soccer ball. In the South African context, functions can be used to model population growth, electricity consumption, the spread of diseases, and even financial planning. A solid understanding of functions is therefore essential for future studies in mathematics, science, engineering, and economics, as well as for informed decision-making in everyday life.

Lesson notes

2.1 What is a Function? A relation is a set of ordered pairs (x, y). We often represent relations using equations, tables, graphs, or mapping diagrams. A function is a special type of relation where each x-value (input) is associated with exactly one y-value (output).

Think of it as a machine: you put something in (x), and the machine gives you a specific output (y).

Independent Variable (x): The input value. We choose the x-value.

Dependent Variable (y): The output value. The y-value depends on the x-value we choose.

Domain: The set of all possible x-values (inputs) for which the function is defined.

Range: The set of all possible y-values (outputs) that the function can produce. 2.2 Representing Functions Functions can be represented in several ways: Equation: This is the most common way.

Example: y = 2x + 1 Table: A table shows pairs of x and y values. | x | y | |---|---| | 0 | 1 | | 1 | 3 | | 2 | 5 | Graph: A visual representation of the function on a coordinate plane. Plot the (x, y) pairs.

Mapping Diagram: A visual way to show the relationship between x and y values using arrows.

The Vertical Line Test: A graph represents a function if and only if every vertical line intersects the graph at most once. This test helps visually determine if each x-value has only one y-value. 2.3 Determining Domain and Range Algebraically: Look for restrictions.

Common restrictions include: Division by zero: The denominator cannot be zero.

Square roots of negative numbers: You cannot take the square root of a negative number (in the real number system).

Example: f(x) = 1/x. The domain is all real numbers except x = 0 (because you can't divide by zero). We write this as x ∈ ℝ, x ≠

0. The range is also all real numbers except y=

0. Example: g(x) = √(x - 2). The domain is x ≥ 2 (because x-2 must be non-negative). We write this as x ∈ ℝ, x ≥

2. The range is y ≥

0. Graphically: Domain: Look at the x-values covered by the graph.

Range: Look at the y-values covered by the graph. Use interval notation to express the domain and range. For example, if the graph extends from x = -3 to x = 5 (inclusive), the domain is [-3, 5]. If it extends infinitely to the right, we write [ -3, ∞). 2.4 Function Notation Function notation is a way to name functions and specify input values. Instead of writing "y = 2x + 1", we write "f(x) = 2x + 1".

This means: "f" is the name of the function. "x" is the input variable. "f(x)" is the output value (the same as "y"). To evaluate a function, substitute the given value for x.

Example: If f(x) = 2x + 1, then f(3) = 2(3) + 1 =

7. Example: If g(x) = x 2 - 4, then g(-2) = (-2) 2 - 4 = 4 - 4 =

0. Example: If h(x) = (x + 1)/ (x - 2), then h(5) = (5 + 1) / (5 - 2) = 6/3 = 2. 2.5 Basic Linear and Quadratic Functions Linear Function: A function of the form f(x) = mx + c, where m is the gradient and c is the y-intercept. The graph is a straight line. To sketch the graph, find two points (e.g., x-intercept and y-intercept) and draw a line through them.

Quadratic Function: A function of the form f(x) = ax 2 + bx + c, where a ≠

0. The graph is a parabola.

Key features: Turning point (vertex): The minimum or maximum point of the parabola. The x-coordinate of the turning point is given by x = -b / 2a.

Axis of symmetry: A vertical line passing through the turning point, dividing the parabola into two symmetrical halves. Its equation is x = -b / 2a. x-intercepts: The points where the parabola crosses the x-axis (where f(x) = 0). y-intercept: The point where the parabola crosses the y-axis (where x = 0).