Functions #2 - Questions and Answers

Question 1 Find an expression for the function whose graph is the given curve. The line segment joining the points (-5, 10) and (7, 10) Answer 1 Equation of the line passing through (x₁, y₁) and (x₂, y₂) is (y - y₁)/(x - x₁) = (y₂ - y₁)/(x₂ - x₁) Equation of the line passing through (-5, 10) and (7, 10) is (y - 10)/(x - (-5)) = (10 - 10)/(7 - (-5)) (y - 10)/(x + 5) = -20/12 (y - 5)/(x + 5) = 5/3 Multiply both sides by x+5 y - 10 = (-5/3)x + (-25/3) Add 10 to both sides y - 10 + 10 = (-5/3)x -25/3 + 30/3 y = (-5/3)x + 5/3 Note that is this equation of the line, to get the equation of the line segment include the limits on x y = (-5/3)x + 5/3 Answer 2 To find an expression for the function whose graph is the line segment joining the points (-5,10) and (7,10), we can start by noting that the function is a horizontal line since the y- coordinates of both points are the same (10). Let's denote the function as f(x). Since the line is horizontal, the value of f(x) is constant for all values of x. In this case, f(x) = 10 for any x within the interval [-5, 7]. Therefore, the expression for the function can be written as: f(x) = 10 Answer 3 Step 1: The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula: m= (y₂ - y₁) / (x₂ - x₁) Using the coordinates of the given points (-5, 10) and (7, 10), we can calculate the slope: m= (10 - 10) / (7 - (-5)) = 0/12 = 0 Step 2: Since the slope is 0, we know that the line is a horizontal line. A horizontal line has the same y-coordinate for all points along its length. Therefore, the equation of the line can be written as: y = c where c is a constant representing the y-coordinate of any point on the line. In this case, since the y-coordinate is always 10, we can rewrite the equation as: y = 10 Thus, the expression for the function whose graph is the given line segment is y = 10. Answer 4 Result: y = 10 Solution: We'll use the point-slope form of the equation of a line, which is given by: y - y₁ = m(x - x₁) where (x₁,y₁) are the coordinates of one point on the line, and m is the slope of the line. First, let's calculate the slope of the line using the formula: m= (y₂ - y₁) / (x₂ - x₁) In this case, (x₁,y₁) = (-5,10) and (x₂, y₂) = (7,10). Substituting these values into the slope formula, we get: m= (10 - 10) / (7 - (-5)) = 0/12 = 0 Since the slope of the line is zero, the line is horizontal. We know that any horizontal line can be expressed as y = c, where c is a constant. In this case, the constant value is 10, as it represents the y-coordinate of the line. Therefore, the expression for the function whose graph is the line segment joining the points (-5, 10) and (7, 10) can be written as: y = 10 The graph of this function is a horizontal line at the y-coordinate 10, as desired. Question 2 The price p (in dollars) and the quantity x sold of a certain product obey the demand equation p = -1/6x + 100. Find a model that expresses the revenue R as a function of x. (Remember, R=xp.) Answer R = xp R= x(-1/6x + 100) R= -1/6x² + 100x Answer is: R = -1/6x² + 100x Question 3 Find a solution of x dy/dx = y² - y that passes through the indicated points. a. (0,0) b. (1/4, 1/4) c. (6, 1/8) Answer 1 By separating variables in x dy/dx = y² - y, we have dy/(y²-y) = dx/x dy/(y(y-1)) = dx/x (-1/y + 1/(y-1))dy = dx/x On integration, we obtain -ln(y) + ln(y-1) = ln(x) + lnc ln((y-1)/y) = ln(cx) (y-1)/y = cx y - 1 = yxc y - yxc = 1 y(1 - xc) = 1 y = 1/(1-xc) y = 1/(1 - cx) is a solution and y = 0 is another constant solution a) At (0,0) y = 1/(1-xc) 0 = 1/(1-0) 0 = 1 This is not possible Therefore in this case there is no solution. Answer 2 c) At (6, 1/8) Here, x = 6, y = 1/8 1/8 = 1/(1-6c) 1 - 6c = 8 Substitute c = -7/6 in y = 1/(1-cx) Thus, y = 1 / (1 - (-7/6)x) y = 1 / (1 + 7x/6) y = 6/(6 + 7x) The solution of the initial value problem is y = 6/(6 + 7x) Question 4 The circle whose equation is x² + y² = 625 is where the point E(-7, 24) is located. What is the sine of an angle when it is drawn in standard position and its terminal ray passes through E?. (1)-7 (2) 7/24 (3)-24 (4) 24/25 Answer An angle passing through (x,y) has the sine given by: sine = y/r From point E, y = 24. Since r = √(x² + y²), then r = √625 = 25 so: sine = 24/25 Question 5 Which relation does not represent a function? A. (0,8), (3, 8), (1, 6) B. (4, 2), (6, 1), (8, 9) C. (1, 20), (2, 23), (9, 26) D. (0,3), (2, 3), (2, 0) Answer If there is precisely one output for each input, a relation is a function. Because there are two possible outcomes for the input 2, namely 3 and 0, option D does not represent a function. 16.3s