Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Define a function and use the vertical line test.
• Determine the domain and range of functions.
• Represent functions as equations, tables, and graphs.
• Use function notation like f(x).
• Sketch basic graphs of functions.

Lesson summary

Welcome to Grade 11 Mathematics! This year, we delve deeper into the fascinating world of functions. Functions are not just abstract mathematical concepts; they are powerful tools that help us understand and model real-world relationships. From calculating your monthly cell phone bill to predicting the growth of a business, functions are everywhere. In South Africa, understanding functions is crucial for analyzing economic trends, modeling population growth, and even designing efficient infrastructure. This week, our focus is on revisiting and solidifying your understanding of fundamental function concepts, including domain, range, notation, and graphical representations.

Lesson notes

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

Think of it like a vending machine: you put in a specific amount of money (the input), and you get a specific snack (the output). You wouldn't expect to put in the same amount of money and get two different snacks! Representing Functions Functions can be represented in several ways: Equations: A mathematical rule that defines the relationship between x and y. For example, `y = 2x + 1` or `f(x) = x²`.

Tables: A table of values that shows specific input-output pairs.

Graphs: A visual representation of the function, plotted on a coordinate plane.

Mappings: A visual showing the mapping of inputs (x) to outputs (y). Function Notation We use function notation to represent functions concisely. Instead of writing `y = 2x + 1`, we can write `f(x) = 2x + 1`. The notation `f(x)` means "the value of the function f at x". So, if we want to find the value of the function when x = 3, we write `f(3) = 2(3) + 1 = 7`.

Domain and Range Domain: The set of all possible input values (x-values) for which the function is defined. In other words, what x-values are "allowed"?

Range: The set of all possible output values (y-values) that the function can produce.

Example 1: Determining Domain and Range (Algebraically) Consider the function `f(x) = 3x - 5`.

Domain: Since we can substitute any real number for x in this equation without any issues (no division by zero, no square root of a negative number), the domain is all real numbers.

We write this as: `x ∈ ℝ` (x is an element of the set of real numbers).

Range: Because the function is a linear function with a non-zero gradient, it can produce any real number as an output.

Therefore, the range is also all real numbers.

We write this as: `f(x) ∈ ℝ` or `y ∈ ℝ`.

Example 2: Determining Domain and Range (Graphically) Imagine a parabola opening upwards. Its vertex is at the point (2, -3).

Domain: The x-values extend infinitely to the left and right. Thus, the domain is `x ∈ ℝ`.

Range: The lowest y-value is -3, and the parabola extends upwards infinitely.

Therefore, the range is `y ≥ -3`.

Example 3: Identifying Functions - The Vertical Line Test The vertical line test is a visual method to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, then the graph does not represent a function. This is because, at that x-value, there would be more than one corresponding y-value, violating the definition of a function. Why is understanding domain crucial? Consider the function that calculates the number of taxis you can hire given a certain budget. The domain would not be all real numbers. It would only be non-negative integers, as you cannot hire a fraction of a taxi, and you cannot spend a negative amount of money.

Example 4: The Hyperbola Consider the function `f(x) = 2/x`.

Domain: The denominator cannot be zero.

Therefore, x cannot be

0. We write this as `x ∈ ℝ, x ≠ 0`.

Range: Since x can be any number except 0, f(x) can be any number except

0. We write this as `f(x) ∈ ℝ, f(x) ≠ 0` or `y ∈ ℝ, y ≠ 0`. The hyperbola has asymptotes at x=0 and y=

0. Guided Practice (With Solutions)

Question 1: Determine whether the following relation represents a function: `{(1, 2), (2, 4), (3, 6), (1, 8)}`.

Solution: This relation does not represent a function because the input value `x = 1` is associated with two different output values, `y = 2` and `y = 8`. This violates the definition of a function.

Question 2: Determine the domain and range of the function `g(x) = x² + 1`.

Solution: Domain: We can square any real number, so the domain is all real numbers: `x ∈ ℝ`.

Range: Since `x²` is always non-negative (greater than or equal to zero), the smallest value of `x² + 1` is 1 (when x = 0). The function can produce any value greater than or equal to

1. Therefore, the range is `g(x) ≥ 1` or `y ≥ 1`.

Question 3: Given the function `h(x) = 5x - 2`, find `h(4)`.

Solution: To find `h(4)`, we substitute `x = 4` into the function: `h(4) = 5(4) - 2 = 20 - 2 = 18`.

Therefore, `h(4) = 18`.

Question 4: Sketch the graph of `y = x + 2`. Determine its domain and range.

Solution: This is a linear function with a y-intercept of 2 and a gradient of

1. We can find two points to plot the line, for example (0,2) and (-2,0).

Domain: All real numbers: `x ∈ ℝ`.

Range: All real numbers: `y ∈ ℝ`.

Question 5: Sketch the graph of `y = x^2`. Determine its domain and range.

Solution: This is a parabola with its vertex at (0,0), opening upwards.

Domain: All real numbers: `x ∈ ℝ`.

Range: `y ≥ 0`. Independent Practice (Questions Only) Determine whether the following relation represents a function: `{(2, 3), (4, 5), (6, 7), (8, 9)}`. Determine the domain and range of the function `f(x) = √x` (the square root of x). Given the function `k(x) = -2x + 7`, find `k(-1)` and `k(0)`. Determine the domain and range of the function `p(x) = 1/(x-1)`.