Mathematics Answer Note #29

Question 1 Operate the ff function f(×) = 2x² + 3x + 1 g(×) = ׳ + 1 b. ( f • g ) ( -1 ) Answer To start off, it is important to understand what the notation "f • g" means. It stands for the composition of two functions, in which the output of the first function becomes the input of the second function. In mathematical terms, it can be written as (f • g)(x) = f(g(x)). Now, let's apply this concept to the given question. In order to find the value of (f • g)(-1), we first need to find the values of f(-1) and g(-1). To do so, we will substitute -1 for x in the respective functions. f(-1) = 2(-1)² + 3(-1) + 1 [substituting -1 for x] f(-1) = 2(1) - 3 + 1 f(-1) = 2 - 3 + 1 f(-1) = 0 g(-1) = (-1)³ + 1 [substituting -1 for x] g(-1) = -1 + 1 g(-1) = 0 Now, we can plug in the values of f(-1) and g(-1) into (f • g)(x) = f(g(x)). (f • g)(x) = f(g(x)) (f • g)(-1) = f(g(-1)) [substituting -1 for x] (f • g)(-1) = f(0) [since g(-1) = 0 as shown above] (f • g)(-1) = 2(0)² + 3(0) + 1 [substituting 0 for x] (f • g)(-1) = 0 + 0 + 1 (f • g)(-1) = 1 Therefore, the value of (f • g)(-1) is 1. Question 2 you need to calculate the points earned by a soccer team the team won 18 games and ended 7 games as a draw a win brings 3 points while draw brings 1 coding program Answer Calculating points in a soccer league is an important aspect of analyzing a team's performance. It determines the team's position in the league standings and their chance of winning the league. The number of points earned by a team is directly related to their wins, losses, and draws. In this scenario, the team has won 18 games and ended 7 games as a draw. To calculate the points earned by the team, we will use the following formula: Total points = (Number of wins * 3) + (Number of draws * 1) Using the values from the given scenario, we can calculate the points earned by the team: Total points = (18 * 3) + (7 * 1) = 54 + 7 = 61 points Therefore, the team has earned 61 points in the league standings. Question 3 What is the value of 4 squared? 4 8 16 64 Answer The value of 4 squared is 16. To understand the value of 4 squared, it is helpful to revisit the concept of exponents. An exponent is a numerical notation indicating the number of times a quantity is multiplied by itself. For example, 4 squared, written as 4^2, means that the number 4 is multiplied by itself 2 times, resulting in 16. Another way to show this is 4 x 4 = 16. In mathematics, the concept of squaring a number is important because it allows us to simplify and solve equations more efficiently. It is also used in real-world applications such as measuring area and volume. One can also think of squaring a number as finding the area of a square with side length equal to that number. In this case, the side length would be 4, and the area would be 16 square units. In conclusion, the value of 4 squared is 16, which can be easily understood by revisiting the concept of exponents. The uses of this mathematical operation are diverse and can be applied both in academic and real-world scenarios. For further understanding, the provided sources can be consulted. Question 4 The numerator of an irreducible fraction is one less than the denominator. If we add the reciprocal fraction to this fraction, the value of their sum will be 113. Find this fraction. Answer To solve this problem, we will first define what an irreducible fraction is. An irreducible fraction is a fraction in which the numerator and denominator have no common factors other than 1. This means that the fraction cannot be simplified any further. To begin solving this problem, let's represent the unknown fraction as x. We can then write the given information in the form of equations: Numerator of x: x - 1 Denominator of x: x Reciprocal of x: 1/x Sum of x and its reciprocal: x + 1/x = 113 We can now use algebraic manipulation to solve for the unknown fraction. Multiplying both sides by x, we get: x^2 + 1 = 113x Subtracting 113x from both sides, we get: x^2 - 113x + 1 = 0 Using the quadratic formula, we can solve for x: x = (113 ± √(113^2 - 4(1)(1))) / 2 x = (113 ± √12769) / 2 x = (113 ± 113) / 2 From the two possible values for x, we can eliminate x = 113 as it does not satisfy the given equation. Thus, the only possible solution for x is x = 1. Therefore, the fraction in question is 1/1 or simply 1. Question 5 A car tyre had its price increased by 8%. The new price is £23.22. What was the original price? Answer In order to find the original price of the car tyre, we can use the following formula: New Price = Original Price + (Original Price * Increase Percentage) Therefore, we can rearrange the formula to find the original price: Original Price = (New Price - (Original Price * Increase Percentage)) / (1 + Increase Percentage) Plugging in the given values, we get: Original Price = (23.22 - (Original Price * 0.08)) / (1 + 0.08) Original Price = (23.22 - 0.08Original Price) / 1.08 Multiplying both sides by 1.08, we get: 1.08Original Price = 23.22 - 0.08Original Price 1.16Original Price = 23.22 Original Price = 23.22 / 1.16 Hence, the original price of the car tyre was £20.05.