Lesson Notes By Weeks and Term v5 - Grade 10
Functions – Week 1 focus
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Functions – Week 1 focus
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Subject: Mathematics
Class: Grade 10
Term: 2nd Term
Week: 1
Theme: General lesson support
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Functions are a fundamental concept in mathematics and play a vital role in understanding relationships between different variables. This week, we'll lay the groundwork for a deep understanding of functions, focusing on their definition, notation, and different ways to represent them. These skills are not just abstract mathematical ideas; they help us model and understand real-world phenomena all around us. For instance, calculating cellphone costs (where call time relates to the total bill), predicting the growth of a business based on investment, or even understanding the spread of information online relies on the concepts of functions.
2. 1.
Relations and Functions: Defining the Difference A relation is simply a set of ordered pairs (x, y). Think of it as a connection between two things. For example, the heights and ages of students in your class.
However, not all relations are functions. A function is a special type of relation where each input (x-value) has exactly one output (y-value). In simpler terms, for every x you put in, you get only one y out. This is crucial. If you have a relation where one x-value gives you two different y-values, it's NOT a function.
Vertical Line Test: A visual way to determine if a graph represents a function is the vertical line test. If any vertical line intersects the graph at more than one point, then the graph does NOT represent a function. This is because that x-value corresponds to multiple y-values. 2.
2. Representing Functions Functions can be represented in several ways: Verbal Description: A function can be described in words. For example, "The output is the square of the input plus 3." Table: A table lists pairs of input and output values. | Input (x) | Output (y) | | --------- | ---------- | | 1 | 4 | | 2 | 7 | | 3 | 12 | Equation: This is a mathematical formula that defines the relationship between the input (x) and the output (y). For example, y = x² + 3 or f(x) = x² +
3. Set of Ordered Pairs: This is a collection of (x, y) pairs. For example, {(1, 4), (2, 7), (3, 12)}.
Graph: A visual representation of the function on a coordinate plane. The x-axis represents the input, and the y-axis represents the output. 2.
3. Function Notation Instead of writing "y", we often use function notation, which looks like this: `f(x)`. `f(x)` means "the value of the function f at x". `x` is the input (independent variable). `f(x)` is the output (dependent variable) – its value depends on what `x` is. `f(2)` means "find the value of the function f when x is 2".
Example: If f(x) = 2x + 1, then f(2) = 2(2) + 1 = 5. 2.
4. Domain and Range The domain of a function is the set of all possible input values (x-values) for which the function is defined. What are all the numbers you are allowed to put into the function? The range of a function is the set of all possible output values (y-values) that the function can produce. What are all the numbers that can come out of the function?
Common Restrictions on the Domain: Division by zero: The denominator of a fraction cannot be zero. Square root of a negative number (in real numbers): You can't take the square root of a negative number and get a real number answer.
Example 1: Cellphone Costs Suppose a cellphone company charges R1.50 per minute of call time. This can be represented by the function C(t) = 1.5t, where C(t) is the total cost and t is the call time in minutes. If you talk for 10 minutes, C(10) = 1.5(10) = R
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5. The domain would be t ≥ 0 (you can't have negative call time). The range would be C(t) ≥ 0 (the cost cannot be negative).
Example 2: The Function f(x) = 1/(x-2)
Equation: f(x) = 1/(x-2) What x-values are allowed? We cannot divide by zero, so x - 2 ≠ 0, which means x ≠
2. Domain: All real numbers except x =
2. We can write this as {x ∈ ℝ | x ≠ 2}.
Range: All real numbers except y =
0. As x approaches 2, the value of f(x) becomes infinitely large (positive or negative). We can write this as {y ∈ ℝ | y ≠ 0}.
Example 3: The Function g(x) = √ (x - 3)
Equation: g(x) = √(x - 3) What x-values are allowed? We cannot take the square root of a negative number, so x - 3 ≥ 0, which means x ≥
3. Domain: All real numbers greater than or equal to
3. We can write this as {x ∈ ℝ | x ≥ 3}.
Range: All real numbers greater than or equal to
0. Since the square root is always non-negative. We can write this as {y ∈ ℝ | y ≥ 0}. Guided Practice (With Solutions)
Question 1: Determine whether the following relation represents a function: {(1, 2), (2, 4), (3, 6), (1, 5)}. Explain why or why not.
Solution: No, this relation is not a function. The input value x = 1 has two different output values: y = 2 and y =
5. This violates the definition of a function.
Question 2: Given the function f(x) = 3x - 2, find f(0), f(2), and f(-1).
Solution: f(0) = 3(0) - 2 = -2 f(2) = 3(2) - 2 = 6 - 2 = 4 f(-1) = 3(-1) - 2 = -3 - 2 = -5 Question 3: Determine the domain of the function h(x) = 5 / (x + 3).
Solution: The function h(x) is defined as long as the denominator is not equal to zero. So, we need to find the values of x for which x + 3 ≠
0. Solving this inequality, we get x ≠ -
3. Therefore, the domain of h(x) is all real numbers except -
3. We can write this as {x ∈ ℝ | x ≠ -3}.
Question 4: A taxi charges a flat rate of R15 plus R8 per kilometer. Write a function that represents the total cost (C) as a function of the distance traveled (d). What is the cost for a 5 km ride?
Solution: The function is C(d) = 15 + 8d. For a 5 km ride, C(5) = 15 + 8(5) = 15 + 40 = R
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5. Independent Practice (Questions Only) Determine whether the following relation represents a function: {(2, 3), (4, 5), (6, 7), (8, 9)}.