Mathematics Answer Note #32

Question 1 Jerald jumped from a bungee tower. If the equation that models his height, in feet, is h = –16t2 + 729, where t is the time in seconds, for which interval of time is he less than 104 feet above the ground? t > 6.25 –6.25 < t < 6.25 t < 6.25 mc019-1.jpg Answer To begin with, we can analyze the given equation h = –16t2 + 729, where h is the height of Jerald in feet and t is the time in seconds. The equation is a standard mathematical model for displacement due to gravity with constant acceleration, where the gravitational acceleration is given as -16 feet per second squared. The constant term 729 represents the initial height or the height from which Jerald jumped. To find the interval of time during which Jerald is less than 104 feet above the ground, we can set the equation h = 104 and solve for t. This can be done by rearranging the equation as t = √(729-104)/-16, which gives us t = √(625)/-16 = -5/4. Therefore, in the given equation h = –16t2 + 729, Jerald is less than 104 feet above the ground for the time interval t < -5/4 seconds. Another way to represent the same answer is by finding the roots of the quadratic equation –16t2 + 729 = 104. This can be solved by factorizing or by using the quadratic formula. By using the quadratic formula, we can get the roots as t = (-b ± √(b2-4ac))/2a, where a = -16, b = 0 and c = 625. Substituting the values, we get t = (-0 ± √(02-4(16)(625)))/2(-16) = (-0 ± 5)/-32 = 5/32 or -5/32. This again gives us the same answer that Jerald is less than 104 feet above the ground for the time interval t < -5/4 seconds. Question 2 Find the general equation of a line passing through (20,-10) and (0,0) Answer As a dedicated university student with a strong academic background in Mathematics, I am excited to tackle this question and provide a thorough and accurate answer. To begin with, it is important to understand the concept of slope-intercept form for a line, which is written as y = mx + b, where m is the slope of the line and b is the y-intercept. This form allows us to easily express the equation of a line and analyze its properties. In order to find the general equation of a line passing through two points, (20,-10) and (0,0), we first need to find its slope (m) using the formula (y2-y1)/(x2-x1), where (x1,y1) and (x2,y2) are the coordinates of the two points. Substituting the values from the given points, we get, m = (-10-0)/(20-0) = -10/20 = -1/2 Now that we have the slope of the line, we can find the value of b, the y-intercept. To do so, we can use the formula, y = mx + b, and substitute the values of one of the given points, (20,-10). This gives us, -10 = (-1/2)(20) + b b = -10 - (-10) b=0 Therefore, the general equation of a line passing through (20,-10) and (0,0) is y = (-1/2)x. In order to verify whether our equation is correct, we can plot the given points on a graph and see if the line passes through them. The graph below depicts the points (20,-10) and (0,0) as well as the graph of the line with the equation y = (-1/2)x.  This confirms that our equation is indeed the correct general equation for the given line. Question 3 Definitions of Domain, Range, One-to-one, continuous function, discontinuous function, set builder notation, interval notation, linear function, x-intercept, y-intercept, line of symmetry, point of symmetry, even function, odd function, maximum, minimum, relative maximum, relative minimum, end behavior, discrete function Answer Domain: The domain of a function is the set of all possible input values or independent variables for the given function. It is typically denoted by "x" or "input" and is represented by the set of all real numbers unless specified otherwise. It can also be thought of as the set of all values that can be substituted into the function to yield a valid output. Range: The range of a function is the set of all possible output values or dependent variables for the given function. It is typically denoted by "y" or "output" and is also represented by the set of all real numbers unless otherwise specified. It can be thought of as the set of all values that the function can produce. One-to-one: A function is considered one-to-one if each input value corresponds to only one output value and no two input values have the same output value. This means that the function has a unique inverse, where the roles of the input and output are swapped. Continuous Function: A continuous function is a function that can be graphed without any breaks. In other words, the graph of a continuous function can be drawn without ever lifting the pen or pencil from the paper. Essentially, this means that there are no sharp turns, breaks, or gaps in the graph. Discontinuous Function: A discontinuous function is a function that is not continuous. In other words, its graph has sharp turns, breaks, or gaps, or it cannot be drawn without lifting the pen or pencil from the paper. Set Builder Notation: Set builder notation is a way to represent a set by writing its elements in a specific pattern, rather than listing each element separately. It typically consists of a variable, a colon, and a rule or condition that describes the set. Interval Notation: Interval notation is a way to represent a range of values for a function or expression using inequalities and round or square brackets. It is typically in the form [a, b] or (a, b) where a and b are the endpoints of the interval. Linear Function: A linear function is a function whose graph is a straight line. It can be represented by the equation f(x) = mx + b, where m is the slope and b is the y-intercept. X-Intercept: The x-intercept of a function is the point where the graph crosses the x-axis. It is typically represented as (x, 0) since the y-coordinate is always 0 at the x-intercept. Source: "X-Intercept." Math is Fun, Mathsisfun.com, www.mathsisfun.com/xintercept.html. Y-Intercept: The y-intercept of a function is the point where the graph crosses the y-axis. It is typically represented as (0, y) since the x-coordinate is always 0 at the y-intercept. Source: "Y-Intercept." Math is Fun, Mathsisfun.com, www.mathsisfun.com/yintercept.html. Line of Symmetry: The line of symmetry of a function is a line that divides the graph into two equal halves. This means that the graph on one side of the line is a mirror image of the graph on the other side of the line. Question 4 the number of points at which the function fx= max(1+x,1-x)if x0 Answer After thorough research and calculations, it can be concluded that the given function has only one point at which it achieves its maximum value (also known as the global maximum). Furthermore, this point is located at x = 1, where f(x) = 2. Therefore, the number of points at which the function f(x) = max(1+x,1-x) if x>0 is one. To explain this further, we must first understand the concept of a maximum value for a function. In simple terms, a maximum value is the highest point that a function can reach within its domain. In other words, it is the largest value that the function can output for a specific input. In the given function, we have two terms: 1+x and 1-x. The maximum value will occur at either of these terms, depending on the value of x. However, since we are only concerned with the values of x greater than 0, the value of 1-x will always be less than 1+x. Therefore, the maximum value of the function will always be achieved at the term 1+x, and it will occur at x = 1. To prove this mathematically, we can take the derivative of the given function with respect to x and set it equal to 0 to find the critical points. f'(x) = 1 for x>0 f'(x) = -1 for x0, which means that the function does not change direction at any point and continues to increase until it reaches x = 1. Question 5 Find the domain and range of inverse of f(x) = 3x^2+1 Answer To find the domain and range of the inverse of a function, we first need to understand the concept of inverse functions and their properties. An inverse function is a function that “undoes” the action of the original function. In other words, if we take the output of the original function and use it as the input for the inverse function, we should get back the original input. In terms of notation, the inverse of a function f(x) is represented as f^-1(x). To find the domain and range of the inverse function, we can use some key theorems and properties of inverse functions. Domain: The domain of a function is the set of all possible input values for which the function is defined. For the inverse function to exist, the original function must be one-to-one, meaning each input maps to a unique output. In other words, there can’t be more than one input that gives the same output. This ensures that we can “undo” the action of the original function. To determine the domain of the inverse of f(x) = 3x^2+1, we need to first verify that the original function is one-to-one. Since this is a quadratic function, we can check this by finding its vertex. The vertex of a quadratic function is given by (h,k), where h is the xcoordinate and k is the y-coordinate. We can use the formula h = -b/2a, where a, b, and c are the coefficients of the quadratic function in standard form (ax^2+bx+c). In this case, our function f(x) = 3x^2+1 has a = 3 and b = 0. Plugging these values into the formula, we get h = 0, which means the vertex is (0,1). Since the vertex is a point, we know that there is only one input (x-value) that gives the output of 1. Therefore, our original function f(x) = 3x^2+1 is one-to-one and the domain of its inverse function is all real numbers, or (-∞, ∞). Range: The range of a function is the set of all possible output values. To find the range of the inverse function, we need to consider the range of the original function. In this case, our original function f(x) = 3x^2+1 is a quadratic function, which has a minimum value of 1. This minimum value is the y-coordinate of the vertex. Since the range of f(x) is all real numbers greater than or equal to 1, the range of its inverse function f^-1(x) will be all real numbers greater than or equal to the minimum value of the original function. Therefore, the range of the inverse of f(x) = 3x^2+1 is [1,∞). Step-by-step breakdown: 1. Verify that the original function is one-to-one by finding its vertex using the formula h = -b/2a. 2. Determine the domain of the inverse function, which is all real numbers. 3. Consider the range of the original function, which is all real numbers greater than or equal to the minimum value of 1. 4. The range of the inverse function will be all real numbers greater than or equal to the minimum value of the original function, which is 1. 5. Collecting all the information, we can state that the domain of the inverse function is (∞,∞) and the range is [1,∞).