Functions Review Packet from November Questions. The diagrams below show the graphs of two functions, = f(), and = g()..5 = f( ) = g( ).5 6º 8º.5 8º 6º.5 State the domain and range of the function f; the function g. (Total 8 marks). f : 5 is a mapping from the set S to the set T as shown below. S T q 5 p 7 Find the values of p and q. () A function g is such that g () =. ( ) (i) (ii) (iii) State the domain of the function g(). State the range of the function g(). Write down the equation of the vertical asmptote. () () () (Total 6 marks
. The following curves are sketches of the graphs of the functions given below, but in a different order. Using our graphic displa calculator, match the equations to the curves, writing our answers in the table below. (the diagrams are not to scale) A B C D E F Function Graph label (i) = + (ii) = + (iii) = (iv) = + (v) = + (vi) = 8
. sin() Sketch the graph of the function =+ for 6 on the aes below. () 9 8 7 6 Write down the period of the function. Write down the amplitude of the function. () () 6. Sketch the graph of = + for. Write down the equations of (i) (ii) the horizontal asmptote; the vertical asmptote.
5. The figure below shows the graphs of the functions = and = for values of between and 5. The points of intersection of the two curves are labelled as B, C and D. 5 C 5 B D A 5 Write down the coordinates of the point A. Write down the coordinates of the points B and C. Find the -coordinate of the point D. (d) Write down, using interval notation, all values of for which. () () () () (Total 8 marks) 7. The graph of a quadratic function f () intersects the horizontal ais at (, ) and the equation of the ais of smmetr is =. Write down the -coordinate of the other point where the graph of = f () intersects the horizontal ais. = f () reaches its maimum value at = 5. (i) (ii) Write down the value of f ( ). Find the range of the function = f ().
8. The figure below shows the graphs of the functions f () = +.5 and g () = for values of between and. 6 f () B A g () Write down the coordinates of the points A and B. Write down the set of values of for which f () g (). 9. The temperature ( C) during a hour period in a certain cit can be modelled b the function T (t) = sin (bt) +, where b is a constant, t is the time in hours and bt is measured in degrees. The graph of this function is illustrated below. T ( C) 5 t (hours) Determine how man times the temperature is eactl C during this hour period. Write down the time at which the temperature reaches its maimum value. Write down the interval of time in which the temperature changes from C to C. (d) Calculate the value of b. 5
. A function is represented b the equation f () = () +. The table of values of f () is given below. f ().75.75 a 7 b Calculate the values for a and b. () On graph paper, draw the graph of f (), for, taking cm to represent unit on both aes. () The domain of the function f () is the real numbers,. Write down the range of f (). () (d) Using our graph, or otherwise, find the approimate value for when f () =. () (Total marks). Below is a graph of the function = a + b sin where a, b and c are positive integers and is measured in degrees. 7 6 5 5 5 5 5 Find the values of a, b and c. 6
. The functions f and g are defined b f :,, g :, Sketch the graph of f for. Write down the equations of the horizontal and vertical asmptotes of the function f. () () Sketch the graph of g on the same aes. () (d) Hence, or otherwise, find the solutions of =. (e) Write down the range of function f. () (). Sketch the graph of the function f : + sin, where, 6 6. () Write down the range of this function for the given domain. Write down the amplitude of this function. () () (d) On the same diagram sketch the graph of the function g : sin, where, 6 6. ( (e) (f) Write down the period of this function. Use the sketch graphs drawn to find the number of solutions to the equation f () = g () in the given domain. () () (g) Hence solve the equation + sin = sin for 6. () (Total 7 marks) 7
. f The diagram shows a function f, mapping members of set A to members of set B. (i) Using set notation, write down all members of the domain of f. A 5 9 6 B (ii) Using set notation, write down all members of the range of f. (iii) Write down the equation of the function f. The equation of a function g is g() = +. The domain of g is. Write down the range of g. 5. Sketch a graph of = for. Hence write down the equations of the horizontal and vertical asmptotes. 6. Two functions are defined as follows f () = g () = 6 for 6 6 for 6 Draw the graphs of the functions f and g in the interval, 8 using a scale of cm to represent unit on both aes. (5) (i) Mark the intersection points A and B of f () and g () on the graph. (ii) Write down the coordinates of A and B. Calculate the midpoint M of the line AB. () () (d) Find the equation of the straight line which joins the points M and N. () (Total marks) 8
7. The diagram shows the graph of = sin a + b. 6 9 5 8 7 6 Using the graph, write down the following values (i) (ii) the period; the amplitude; (iii) b. Calculate the value of a. (Total 8 marks) 8. Write down the domain and range of the following functions. (Total 8 marks) 9
9. The area, A m, of a fast growing plant is measured at noon (:) each da. On 7 Jul the area was m. Ever da the plant grew b 7.5%. The formula for A is given b A = (.75) t where t is the number of das after 7 Jul. (on 7 Jul, t = ) The graph of A = (.75) t is shown below. A 6 6 8 6 8 7 Jul t What does the graph represent when t is negative? () Use the graph to find the value of t when A = 78. () Calculate the area covered b the plant at noon on 8 Jul. (). Consider the function f () = sin where 7. Write down the period of the function. Find the minimum value of the function. Solve f () = l. (Total 8 marks)
. The diagrams below are sketches of some of the following functions. (i) = a (ii) = a (iii) = a (iv) = a (v) = a (d) DIAGRAMS NOT TO SCALE Complete the table to match each sketch to the correct function. Sketch (d) Function (Total 8 marks). The diagram below shows a part of the graph of = a. The graph crosses the -ais at the point P. The point Q (, 6) is on the graph. Q (, 6) P O Diagram not to scale Find the coordinates of the point P; the value of a.
(Total 8 marks). The graph below shows the curve = k( ) + c, where k and c are constants. 6 Find the values of c and k. (Total marks). The diagram below shows the graph of = a sin + c, 6. 5 9 8 7 6 Use the graph to find the values of c; a. (Total marks)
5. The following diagram shows part of the graph of an eponential function f() = a, where. f () P What is the range of f? Write down the coordinates of the point P. What happens to the values of f() as elements in its domain increase in value? (Total marks) 6. A function f is represented b the following mapping diagram. 5 7 Write down the function f in the form f :, {the domain of f}. The function g is defined as follows g : sin 5, { and < }. Complete the following mapping diagram to represent the function g. (Total marks)
7. Diagram shows a part of the graph of =. Diagram Diagrams, and show a part of the graph of = after it has been moved parallel to the -ais, or parallel to the -ais, or parallel to one ais, then the other. Diagram Diagram Diagram Write down the equation of the graph shown in Diagram ; Diagram ; Diagram. (Total marks)