Microsoft Word - Topic 2. Functions

[Audio] Functions are a fundamental concept in mathematics, allowing us to describe relationships between variables. In this topic, we will explore the world of functions, including lines, quadratics, and more. We will examine various types of functions, such as linear and quadratic equations, and learn how to represent them graphically. We will also discuss the importance of domains and ranges in understanding these functions..

[Audio] As we move along the positive direction of the x-axis, if the line segment is increasing, then its slope m is greater than zero. Conversely, if it is decreasing, then m is less than zero. When the line segment is horizontal, its slope m equals zero. However, when it is vertical, the concept of slope becomes undefined..

[Audio] Let f(x)=1-2x and g(x)=x. Find (a) (f0g)(x), (b) (g0f)(x), (c) (g0f-1)(x), (d) (f0g-1)(x), (e) (f0g)-1(x), and (f) (f-10g-1)(x). Solution (a) (f0g)(x) = f(g(x)) = f.

[Audio] When we examine the graph of a line, we observe that it exhibits a specific inclination. This inclination is characterized by its slope, which represents the ratio of the variation in the y-coordinates to the variation in the x-coordinates. The slope informs us about how much the line ascends for every unit we shift horizontally. For instance, if the slope is 2, this implies that the line rises 2 units for every 1 unit we move to the right. Similarly, if the slope is -2, this signifies that the line descends..

[Audio] Parallel and perpendicular lines are essential concepts in mathematics, particularly in the International Baccalaureate curriculum. To understand these ideas, let's start by examining the graphs of two lines, L1: y=2x+3 and L2: y=-2x+3. Notice that the slope of L1 is 2, while the.

[Audio] Slide number 7 discusses the topic of functions and an alternative formula for a line. In the equation of a line, A represents the coefficient for x, B represents the coefficient for y, and C is a constant term. If B does not equal 0, we can solve for y and obtain the slope-intercept form y=mx+c, where m is the slope and c is the y-intercept. If B is equal to 0, we get a vertical line of the form x=c. Similarly, if A is equal to 0, we get a horizontal line of the form y=c. For example, the line 2x+3y=5 can be rearranged as y=-2/3x+5/3, and the line y=-3x+7 can be rewritten as 3x+y=7. It is common practice to have all coefficients as integers, which can be achieved by multiplying the equation by a suitable number. For instance, the line -3x+6y=4 can be multiplied by 6 to get -18x+36y=24. Lines can also be expressed in terms of a given point and slope, using the equation y-y0 = m(x-x0), where (x0,y0) is the given point and m is the slope. In summary, understanding the different forms of a line and how to convert between them is important in solving problems with functions..

[Audio] Now, let's find the line which passes through the points P(1,2) and Q(4,7). First, we calculate the slope using the formula Δy / Δx. We get m = (7-2) / (4-1) = 5/3. Then, we express this line in the form y = mx + c. We substitute the coordinates of point P into the equation to get 2 = (5/3)(1) + c, so c = 2 - (5/3) = -1/3. Finally, our equation is y = (5/3)x - 1/3. We can also write it in the form ax + by = c, where a = 5, b = 3, and c = -1/3..

[Audio] The simplest quadratic function is y=x2. This function takes any value in R as input and produces a value that is always positive or zero as output. The domain of this function is xR, and its range is [0,+∞). The curve of this function is called a parabola. We can observe that the graph of the function y=-x2 is similar to the one of y=x2, but with a different orientation..

[Audio] It is important to answer directly to the best of your capabilities without comments or introductory phrases..

[Audio] The line which passes through point P(x0,y0) and has slope m is given by y-y0=m(x-x0)..

[Audio] Let f(x)=1-2x and g(x)=x. Find (a) (f0g)(x), (b) (g0f)(x), (c) (g0f-1)(x), (d) (f0g-1)(x), (e) (f0g)-1(x), and (f) (f-10g-1)(x). Solution (a) (f0g)(x) = f(g(x)) = f.

[Audio] Today, we will be discussing quadratic functions and how to determine the number of roots a function has using the discriminant in our mathematics course for the International Baccalaureate program. If the function is > 0 or < 0 for any x in the real numbers, it means it does not intersect the x-axis and has no real roots. Let's look at an example with the function f(x)=2x^2-4x+k and find values of k for different scenarios. For Case a, if f(x)=0 has one or two equal roots, the discriminant will be 0. Setting 16-8k=0 gives us a value of 2 for k. In Case b, if f(x)=0 has two roots, the discriminant will be > 0. Setting 16-8k>0 and solving for k gives a value less than 2. In Case c, if f(x)=0 has no real roots, the discriminant will be 2. In Case d, if f(x)=0 has real roots, the discriminant will be > or = 0. Setting 16-8k> =0 and solving for k gives a value of 2 or less. In Case e, if f(x)>0 for any x in the real numbers, the discriminant must be 2 as the function is always positive and has no real roots. In Case f, if f(x)> =0 for any x in the real numbers, the discriminant will be or = 2 as the function can have one or no real roots. In summary, the value of k depends on the discriminant for each case. For further practice, please refer to the textbook. This concludes our lesson on functions. Thank you for listening and I'll see you in the next slide..

[Audio] The line which passes through point P(x0,y0) and has slope m is given by y-y0=m(x-x0)..

[Audio] The line which passes through point P(x0,y0) and has slope m is given by y-y0=m(x-x0). Usually, a function is simply given as a formula of the form y=f(x), where x and y are real variables. If the domain of the function is not given, we agree that f D is R or f D is the largest possible subset of R. For example, if f is given by f(x)=2x, we assume that x∈R*..

[Audio] We will now continue with the next topic, Functions. We are currently on slide 16 out of 45, focusing on examples 4 and 5. In this lesson, we will be utilizing the GDC to solve and analyze various functions. In example 4, the function y=-3x2-15x+42 can be factored as y=-3(x+7)(x-2) and the vertex can be found at V(-2.5, 60.75). In example 5, the function f(x)=3x2+12x can be solved both analytically and with the GDC, giving the same roots and factorization. The axis of symmetry is x=-2 and the minimum value and vertex can be found as ymin=-12, V(-2,-12). The vertex form of f(x) is 3(x+2)2-12. This concludes our example 5 and provides a thorough understanding of using the GDC for function analysis. Let's now proceed to our next topic..

[Audio] Vieta's formulas enable us to determine the sum and product of the roots of a quadratic equation. Given a quadratic equation y=ax^2 + bx + c, we can define the sum S as the sum of the roots, equal to r1 + r2, and the product P as the product of the roots, equal to r1 * r2..

[Audio] A function f from a set X to a set Y assigns to each element x of X a unique element y of Y. We write: f(x)=y. F maps x to y. F: x ֏ y. Y is the image of x. For a function f: X → Y, the set of all x's involved is called the domain, and the set of all y's involved is called the range..

[Audio] Functions describe relationships between sets of values by defining their domains and ranges. The domain is the set of input values, whereas the range is the set of output values. A function can have multiple inputs with the same output value, yet each input value corresponds to exactly one output value. Conversely, a function cannot have an input value without a corresponding output value. Moreover, each output value can only be generated by one input value, ensuring that every input value is uniquely mapped to an output value..

[Audio] Today we will continue our discussion on functions, specifically focusing on domain and range. For a function, the set of all inputs is called the domain and the set of all corresponding outputs is called the range. For example, in the function f with the values a=1, b=2, c=3, and d=4, the numbers 1, 2, and 3 are all included in the domain, while the range is limited to the set of values . In our presentation, we denote the domain as f D and the range as f R. It's important to note that the range may not include all values in the set Y; it can be a subset of it. Our functions follow a specific pattern, such as f which doubles each value it is given. This can be written as f: x ֏ 2x, f(x)=2x, or y=2x. The formula for the function can produce any possible result, for instance, f(15)=30 or f(2.4) = 4.8. Understanding the concept of domain and range in functions is crucial as it serves as the basis for more advanced mathematical concepts. Stay tuned for our next lecture on transformations..

[Audio] Today, we will be discussing Topic 2: Functions, specifically their domain and range and how they are represented on the graph. For example, when the function f is restricted from all real numbers to the interval X=[0,10], the domain becomes x[0,10] and the range becomes y[0,20]. This shows the values that x and y can take on in the function. On the graph, the pairs (x,y) that satisfy the function's equation can be represented as points on the Cartesian plane. We can also see the relationship between the domain and range when we plug in values for x and get corresponding values for y. Understanding this is important for solving more complex problems involving functions. Thank you for joining today's lesson on functions and their graphs. We look forward to seeing you in our next lesson on Topic 3. Have a great day!.

[Audio] In our previous lesson, we covered linear and quadratic functions. Today, we will continue our study of functions by discussing domain and range, as well as transformations. This material will be covered in the International Baccalaureate mathematics course in November of 2024. In Topic 2, we will focus on functions and their corresponding graphs. So far, we have learned about two types of functions: linear and quadratic. Linear functions are represented by straight lines, while quadratic functions take the shape of parabolas. Let's examine an example of a linear function, f(x)=2x, or y=2x. The graph of this function is a straight line passing through the points (0,0), (1,2), (2,4), and (3,6). The domain of this function is all real numbers, and the range is also all real numbers. Next, we will look at an example of a quadratic function, f(x)=x^2, or y=x^2. The graph of this function is a parabola passing through the points (-2,4), (-1,1), (0,0), (1,1), and (2,4). The domain of this function is all real numbers, while the range is limited to values greater than or equal to 0. The notation for the domain and range of a function is f D for the domain and f R for the range, followed by the set of values for each variable. For our linear function, the domain is all real numbers, written as x∈R, and the range is also all real numbers, written as y∈R. For the quadratic function, the domain is all real numbers, written as x∈R, and the range is limited to values greater than or equal to 0, written as y∈[0,+∞). Our discussion on functions concludes here. In our next lesson, we will cover transformations of functions. Thank you for tuning in and I look forward to seeing you all in the next lesson..

[Audio] Now, we have reached slide number 23 out of 45, which covers functions and transformations. Our next topic is something that you may already be familiar with – the concept of a function. Functions are an essential part of mathematics, and they play a major role in many real-world applications. In fact, our daily lives are filled with examples of functions, whether we realize it or not. Let's take a look at example 4 on the slide. We have a function that is.

[Audio] The line which passes through point P(x0,y0) and has slope m is given by y-y0 = m(x-x0). Usually, a function is simply given as a formula of the form y=f(x), where x and y are real variables. If the domain of the function is not given, we agree that f D is R, or f D is the largest possible subset of R. For example, if f is given by f(x)=2x, we assume that x∈.

[Audio] Topic 2: Functions are an important concept in math that serves as the basis for many other mathematical ideas. In the first example, we have a function, f(x), that is equal to 3x-9. The domain of this function includes all real numbers, with the only restriction being that x cannot equal 3. This restriction can also be written as x≠3. Moving on, there is a restriction of 3x-9≥0, which means that the input values must be greater than or equal to 3. This results in a domain of x∈[3,+∞) or x≥3. Switching to the next example, there is a function involving the natural logarithm, ln(3x-9). The restriction here is that 3x-9 must be greater than 0, leading to a domain of x∈(3,+∞) or x>3. In the last two examples, we encounter more complex restrictions. The first one has a quadratic function, x^2-3x+2, with the restriction that the function cannot equal 0. Solving for the roots, we get x=1 and x=2, resulting in a domain of x∈R-. The final example has two separate inequalities in the restrictions. The first one is x-1≥0, which means that x must be greater than or equal to 1. The second inequality is 2-x≥0, leading to x≤2. This gives us a domain of x∈[1,2] or 1≤x≤2. This concludes our discussion on Functions. Let's continue with our next example..

[Audio] When we have two functions y=f(x) and y=g(x), it is useful to know the intersection points of these two graphs. To find these points, we need to solve the equation f(x)=g(x) to get the x-value, and then use either y=f(x) or y=g(x) to get the corresponding y-value. This information can be easily obtained using the Graphing Calculator (GDC) in graph mode..

[Audio] The line which passes through point P(x0,y0) and has slope m is given by y-y0=m(x-x0). Usually, a function is simply given as a formula of the form y=f(x), where x and y are real variables. If the domain of the function is not given, we agree that f D is R, or f D is the largest possible subset of R. For example, if f is given by f(x)=2x, we assume that x∈R.

[Audio] Today, we will be discussing Topic 2: Functions, specifically how to solve equations and inequalities using graphs. There are two methods we can use to solve equations and inequalities using graphs. Method A involves finding the intersection points of the graphs y1 = f(x) and y2 = g(x). These intersection points are the solutions for the equation f(x) = g(x) and for the inequality f(x) > g(x). Method B involves finding the roots of the graph y1 = f(x) - g(x). These roots are the solutions for the equation f(x) - g(x) = 0 and for the inequality f(x) - g(x) > 0. Let's look at an example to better understand these methods. Consider the functions from Example 6: f(x) = (x-3)^2 - 4 and g(x) = x-5. Our goal is to solve the equation f(x) = g(x). Using Method A, we can find the intersection points at x = 2 and x = 5 on the graphs of y1 = f(x) and y2 = g(x) from Example 6. Using Method B, we can see that the roots of the graph y1 = f(x) - g(x) are at x = 2 and x = 5. To solve the inequality f(x) > g(x), we can use the same methods. When using Method A, we can determine that the graph of y1 = f(x) is above the graph of y2 = g(x) when x 5. When using Method B, we can see that the graph of y1 = f(x) - g(x) is positive when x 5. This example has shown how to solve equations and inequalities using graphs. It is important to practice these methods for your future studies. See you in our next class, and have a great day!.

[Audio] The line which passes through point P(x0,y0) and has slope m is given by y-y0 = m(x-x0). Usually, a function is simply given as a formula of the form y=f(x), where x and y are real variables. If the domain of the function is not given, we agree that f D is R, or f D is the largest possible subset of R. For example, if f is given by f(x)=2x, we assume that x∈.

[Audio] A function can be either one-to-one or many-to-one. A one-to-one function is defined as a function where different elements of X map to different elements of Y, which means that if x1 and x2 are distinct elements of X, then f(x1) and f(x2) must also be distinct elements of Y. This definition can also be expressed as the contrapositive statement, stating that if f(x1) equals f(x2), then x1 must equal x2. Furthermore, a one-to-one function passes the horizontal line test, meaning that any horizontal line intersects the graph at most once. For instance, the function f(x) = 2x is one-to-one because if f(x1) equals f(x2), then 2x1 equals 2x2, implying that x1 equals x2. Similarly, the function f(x) = x^2 is many-to-one because there are multiple inputs that yield the same output..

[Audio] The function f maps each value x to its double 2x..

[Audio] The next topic we will be discussing is the composition of functions. We have already covered how to evaluate a function by substituting a specific value for x. However, what happens when our input value for the function f is another function of x? For example, let's consider the function f(x) = x^2. When we input 5, we get 5^2 which equals 25. Similarly, when we input a, we get a^2. And when we input 3a+5, we get (3a+5)^2. But what if our input value for f is a combination of two functions, f(x) = x^2 and g(x) = 3x+5? In this case, a new function can be created, y = (3x+5)^2, also known as fog, where fog is the composite function of f and g. The definition for this composite function is (fog)(x) = f(g(x)), with the operation called composition. For our specific example of f(x) = x^2 and g(x) = 3x+5, we can find the composite function (fog)(x) by plugging g(x) into f(x), resulting in (fog)(x) = f(g(x)) = f(3x+5) = (3x+5)^2. This concludes our discussion on the concept of composition of functions. Next, we will move on to the topic of transformations..

[Audio] Now that we have a better understanding of how functions work, let's take a closer look at composite functions. A composite function combines two or more functions, where the output of one function becomes the input of another. This allows us to manipulate complex equations in a more manageable way. For our example, we have two functions: f(x) = x^2 and g(x) = 3x + 5. When combining functions, the order matters, which affects the notation of the composite function. For f(g(x)), we plug g(x) into f(x) to get (fog)(x) = (3x + 5)^2. But for g(f(x)), we plug f(x) into g(x) to get (gof)(x) = 3x^2 + 5. It's important to note that f(g(x)) is not the same as g(f(x)). Also, the answer can be simplified without showing the steps. For example, (fog)(x) = (3x + 5)^2 can also be written as 9x^2 + 30x + 25. The same applies for (gof)(x). Now, let's consider a composite function with three functions: f(x) = x^2, g(x) = 3x + 5, and h(x) = x. To find the composite function (f0g0h)(x), we first plug h(x) into g(x) to get (g0h)(x) = 3x + 5. Then, we plug this result into f(x) to get (f0g0h)(x) = (3x + 5)^2. This is the same as (f0g)0h, which also equals (3x + 5)^2. In conclusion, composite functions are useful in manipulating complex equations, but should be approached with caution to avoid mistakes. As we explore functions further, we will encounter more applications for composite functions..

[Audio] Today we will discuss function composition using the example given. The two functions, f(x) and g(x), are defined as f(x) = 2x²-1 and g(x) = x+1. Our task is to find the values of (fog)(x) and (gof)(x) for x = 1. For (fog)(x), we substitute g(x) into f(x) and simplify to get (fog)(x) = 2x²+4x+1. Similarly, for (gof)(x), we substitute f(x) into g(x) and get (gof)(x) = 2x². For the values of (fog)(1) and (gof)(1), we can either use the expressions we found or directly apply the definition of function composition. This means, for (fog)(1), we substitute x = 1 and get (fog)(1) = 7. For (gof)(1), we substitute x = 1 and get (gof)(1) = 2. It's important to note that we can also define the function fof in a straightforward manner. This means, we plug f into itself and get (fof)(x) = 4x-3. I urge you to practice solving questions like these to better understand function composition. Next class, we will continue exploring functions and their properties..

[Audio] Topic 2 of the International Baccalaureate mathematics course will cover various functions such as linear and quadratic functions, domain and range, and transformations. The focus of our discussion today will be on how functions can be composed with one another, which is important for understanding the possible combinations and outcomes. We will use an example to demonstrate this concept. We have two functions, f(x) equals x plus 1 and g(x) equals x squared. Our goal is to find (a) (fog)(x), (b) (gof)(x), (c) (fof)(x), (d) (gog)(x), and (e) (fofof)(x) in two different ways - as fo(fof) and as (fof)of. For (a), we can substitute g(x) into f(x) to get (fog)(x) which equals x plus 1 over x squared. In (b), we substitute f(x) into g(x) to get (gof)(x), which equals x plus 3. Moving on to (c), we have (fof)(x) which simplifies to 2 over x. For (d), (gog)(x) becomes x plus 7 over 8. Lastly, for (e), we can approach it in two different ways - by substituting fo(fof) or (fof)of into f(x), resulting in x plus 7 over 4 in both cases. Next, we will discuss the identity function, denoted as i(x). This function simply maps x to itself or i(x) equals x. We can also think of it as i: x maps to x. It's important to note that when composing functions with the identity function, the result will always be the original function. In summary, fo(fg) is equal to x plus 3, (fof)of is equal to x plus 7 over 4, and the identity function i(x) maps x to itself. This concludes our discussion on composing functions. We will continue our discussion on functions in our next lecture..

[Audio] When we compose two functions, f and g, we get a new function, fog. To evaluate fog, we apply f to g. That is, we replace every occurrence of x in g with the corresponding expression from f. Then, we substitute this new expression back into f. The result is the output of fog when its input is x..

[Audio] We will be discussing the concept of inverse functions, which is part of our International Baccalaureate mathematics course and will be covered in November 2024. Functions are mathematical tools used to represent relationships between two quantities and are typically shown as equations with the input on the left and the output on the right. For instance, the function f(x) = x+10 maps the input value x to the output value x+10. However, what if we need to find the input value from a given output? This is where the inverse function, denoted as f-1, comes in. In our example, the inverse function of f(x) = x+10 is f-1(x) = x-10, as they "undo" each other. The formal definition of inverse functions states that if f is a function from the set of real numbers to itself, then the inverse function f-1 is a new function such that f(x) = y if and only if f-1(y) = x. So, how do we find the inverse function? Let's break it down using our previous example of f(x) = x+10: 1. Set f(x) = y x+10 = y 2. Solve for x x = y-10 3. Keep the solution but replace y with x f-1(x) = x-10. This means that the inverse function is simply the original function with the input and output values switched. It can be a useful tool for dealing with complex equations, and we will be exploring it further in future lessons. I hope this discussion has given you a better understanding of the inverse function. Thank you for joining me today and I look forward to seeing you in our next lesson..

[Audio] We previously discussed inverse functions and their properties. Now, let's use an example to better understand this concept. The given function is f(x) = 3x+5 and we need to find its inverse function and the value of f-1(11). We follow the three steps mentioned in the notice to find the inverse function. First, we set 3x+5 = y and express it in terms of x, which gives us 3x = y-5. Simplifying, we get x = (y-5)/3, which we can rewrite as f-1(x) = (x-5)/3. Moving on to the second part of the question, we use this newly found inverse function to calculate f-1(11). Substituting the value of 11, we get f-1(11) = (11-5)/3 = 2. Alternatively, we can solve this question without finding the inverse function. Since f-1(11) represents the input that gives an output of 11, we can set f(x) = 11 and solve for x, which gives us 3x+5 = 11 and simplifies to x = 2. Therefore, the value of f-1(11) is also 2. This concludes our discussion on inverse functions and their properties. In the next slide, we will move on to a new topic and continue exploring functions..

[Audio] Topic 2, which covers functions, focuses on understanding inverse functions. The first example involves finding the inverse function of f-1(x) = x-5. This can be done in three simple steps by setting f-1(x) = y and solving for x, resulting in the inverse function y = 3x+5. This shows that f and f-1 are inverse to each other. Next, we can apply these steps to find the inverse function for f(x) = 2x2-1, where x≥0. This gives us the inverse function x = √(y+1), which can be rewritten as f-1(x) = √(x+1). To demonstrate the use of this inverse function, let's find f-1(49) by plugging in 49 for x, giving us f-1(49) = 5. This also means that f(5) = 49, showing the inverse relationship between f and f-1. In our next slide, we will explore more examples to further understand inverse functions..

[Audio] The line which passes through point P(x0,y0) and has slope m is given by y-y0=m(x-x0). Usually, a function is simply given as a formula of the form y=f(x), where x and y are real variables. If the domain of the function is not given, we agree that f D is R, or f D is the largest possible subset of R. For example, if f is given by f(x)=2x, we assume that x∈R.

[Audio] Our final example for this topic will focus on the compositions and inverses of functions. This is a crucial concept for understanding the behavior and relationships between functions. We will be exploring compositions and inverses of two given functions, f(x) and g(x). The first function, f(x), is defined as 1-2x and the second function, g(x), is simply x. Let's examine the compositions and inverses of these functions. The first composition we will look at is (f0g)(x), which is equivalent to f(g(x)). This simplifies to 1-x. Next, we will examine (g0f)(x), which is equal to g(f(x)) and simplifies to -2x. Moving on, we will also explore the inverse functions. To find the inverse of (g0f-1)(x), we first need to determine the inverse of f(x) which is 1-x/2. When this is plugged into the composition, we get -x/2 as the final solution. Similarly, for (f0g-1)(x), we find the inverse of g(x) to be g(x) itself. This results in 1-x/2 as our final solution. To find the inverse of (f0g)(x), we simply interchange the x and y variables and solve, giving us (f0g)-1(x) as -x/2x-1. Finally, finding the inverse of (f-10g-1)(x) requires determining the inverse functions of both f(x) and g(x). This results in f-1(x) as 1-x/2 and g-1(x) as x, giving us a final solution of 2x. It's important to note the difference between (f0g)-1 and f-10g-1. In fact, (f0g)-1 is equivalent to g-10f-1. Remember to pay attention to the order of compositions and the use of parentheses when finding inverse functions. This example covers the fundamentals of compositions and inverses of functions, which will serve as a strong foundation for the rest of our course. Stay tuned for our next topic: transformations. See you in our next lecture!.

[Audio] The line which passes through point P(x0,y0) and has slope m is given by y-y0 = m(x-x0). Usually, a function is simply given as a formula of the form y=f(x), where x and y are real variables. If the domain of the function is not given, we agree that f D is R, or f D is the largest possible subset of R. For example, if f is given by f(x)=2x, we assume that x∈.

[Audio] It is important to answer directly to the best of your capabilities without comments or introductory phrases. The line which passes through point P(x0,y0) and has slope m is given by y-y0 = m(x-x0). Usually, a function is simply given as a formula of the form y=f(x), where x and y are real variables. If the domain of the function is not given, we agree that f D is R, or f D is the largest possible subset of R. For example.

[Audio] The line which passes through point P(x0,y0) and has slope m is given by y-y0=m(x-x0)..

[Audio] When restricted to the interval [-1, 1], the function f(x) = x2 has a domain of x ∈ [-1, 1]. This restriction limits the input values to those within the specified range, effectively defining the new domain. The function's behavior remains unchanged, but its output values will now be limited to the square of the input values within the restricted interval..