Final Examination Spring 2016. Question 5-7

Math 1103 Common Final Examination Spring 2016 university of north California Charlotte shahbaz ahmed June 2024 1 An important Question Q5 The graph of the function y = f (x) is given below.Select the answer choice that represent the graph y = f (x + 2) − 1 1 Solution From the graph,slope of the line through the points (−1, 1), (3, −4) −5 −4−1 = 3+1 = −5 slope= 3−(−1) 4 2 Equation of the line through the point (−1, 1) = (x1 , y1 ) with slope = − 45 y − y1 = slope(x − x1 ) y − 1 = − 54 [x − (−1)] y − 1 = − 54 (x + 1) 4y − 4 = −5x − 5 4y = −5x − 5 + 4 = −5x − 1 y = − 54 x − 14 = f (x) Putting in y = f (x + 2) − 1 y = − 54 (x + 2) − 14 − 1 5 y = − 45 x − 10 4 − 4 5 15 y = −4x − 4 At x=1 20 y = − 45 (1) − 15 4 = − 4 = −5 The point (1, −5) lies on the graph B However at x = −5, y = 2.5 (−5, 2.5) does not lie on any graph. Hence answer is none of the above. Hence answer E.N one of the above Q6 Which of the following has an inverse function on the domain (−∞, ∞) I.f (t) = t4 II. g(t) = 1 + 3t III. h(t) = sin t a. I only b. II only c. III only d. II and III only e. I and II only 4 Solution qp I. f (t) = t t= f (t) Domain = [0, ∞) II g(t) = 1 + 3t t = g(t)−1 3 Domain =(−∞, ∞) III h(t) = sin t t = arcsin h(t) Domain =[−1, 1] answer b.IIonly 3 Q7 If (2, −5) is a point on the graph of f(x),which of the following MUST be on the graph of y = 31 f (x − 1) a.( 23 , −4) b.(1, −5) c.( 13 , −5) d.(1, − 35 e.(3, − 53 ) Solution Since f (2) = −5 Or f (2) = 1 − 3times2 = −5 f(x) may be defined as f (x) = 1 − 3x Hence f (x − 1) = 1 − 3(x − 1) = 1 − 3x + 3 = −3x + 4 y = 31 f (x − 1) = 13 (−3x + 4) = −x + 43 Putting x = 3 y = −3 + 34 = − 53 Hence answer e.(3, − 53 ) 4