September 24 (A and E) or 26 (F), 2014 page 1 Review: functions and function operations We will have a test Monday Sept. 29 (E and F Blocks) or Tuesday Sept 30. (A Block) on the material we ve covered from sections 1.2-1.5. Here is a topic outline that you may find useful as a study checklist: function concept (1.2) o function representations: graph, table, f( formula o domain and range o Vertical Line Test (for whether a graph is a function) o evaluating functions and solving equations using graphs (on paper and on calculator) function properties (1.2) o increasing, decreasing, constant o local maxima and minima, absolute maxima and minima o even and odd symmetry (recognizing graphically and confirming algebraically) o drawing graphs with specified properties piecewise-defined functions (1.3) function operations (1.4) o addition, subtraction, multiplication, and division o composition inverse functions (1.5) o finding an inverse by reversing the steps of a rule o Horizontal Line Test (for whether a graphed function has an inverse) o finding an inverse by exchanging the x and y variables of a graph or table o finding an inverse graphically by reflecting across the line y = x o finding an inverse algebraically Review problems Answer on separate paper except where directed to use the grids on this handout. Check your answers after finishing each numbered problem (answers begin on page 4). Following problems 1 and 2 there are boxes containing an important note to take away from each problem. Make sure you read these boxes, because they explain how the ideas from these problems can be applied to other problems. 1. Graphs of f( and g( are given on the grid. a. Find the solutions of f( = g(. b. Let h( = f( g(. Graph h( on the grid. c. Find the zeros of h(. d. You should have gotten the same answers for part a and part c. Why did this happen?
September 24 (A and E) or 26 (F), 2014 page 2 2. Consider f( = x 3. a. What are the domain and range of f(? Write your answers using the bracket notation for intervals. b. Graph f(. You may use your calculator. c. Graph f 1 (. Do not use your calculator. d. What are the domain and range of f 1 (? Write your answers using the bracket notation for intervals. e. Find a function formula for f 1 (. 3. On the given grid, draw a graph of a function f( having these properties: The domain is [ 7, 7]. The range is [ 4, 4]. The graph has odd symmetry. f( is decreasing in the interval [1, 3]. x = 2 is a zero of f(. f(3) = 4. 4. For all parts of this problem, let f( = a. What is the domain of f(? b. Find the zero(s) of f(. 2x 3. x 1 c. Using graphs on your calculator, solve the equation f( = x + 3. d. Algebraically find a function formula for f 1 (, the inverse function of f(.
September 24 (A and E) or 26 (F), 2014 page 3 5. Consider this piecewise-defined function: 2 x if x 2 f ( 4 if 2 x 2 a. Make an input-output table for f( using integer inputs from 4 to 4. b. Draw the graph of f(. c. What kind of symmetry does the graph of f( have? d. Does f( have an inverse function? Explain why or why not. e. Explain why a function with even symmetry can never have an inverse function. 6. Consider functions f and g given by the following tables: x f( x g( 15 10 15 5 10 5 10 10 5 0 5 5 0 5 0 10 5 10 5 5 10 15 10 10 15 10 15 5 a. Make an input-output table for (f g)(. b. Make an input-output table for ( f g)(. c. Each of these tables was created using a function operation of f and g. Identify each table. x??? x??? 15 2 15 10 10 0.5 10 5 5 0 5 10 0 0.5 0 5 5 2 5 10 10 1.5 10 5 15 2 15 10
September 24 (A and E) or 26 (F), 2014 page 4 Review problem answers 1. a. x = 1 and x = 3 b. see graph at the right c. x = 1 and x = 3 d. If h( = 0, then f( g( = 0 so f( = g(. If f( = g(, then h( = f( g( = 0. So h( = 0 and f( = g( are equivalent and have the same solutions. Takeaway note: What happened in this problem will happen with any pair of functions. You can always solve an equation f( = g( by turning it into f( g( = 0 and finding the zeros of the difference. 2. a. domain: [3, ) range: [0, ) b and c. see graphs below d. domain: [0, ) range: [3, ) e. f 1 ( = x 2 + 3, for x 0 Takeaway note: What happened in this problem will always happen for inverse functions: the domain of f 1 is the same as the range of f the range of f 1 is the same as the domain of f
September 24 (A and E) or 26 (F), 2014 page 5 3. Correct answers vary. Any graph meeting all the requirements is a correct answer. As a partial check: o Any correct graph will have to include these points: ( 3, 4), ( 2, 0), (0, 0), (2, 0), (3, 4). o Any correct graph will have local maxima at x = 3 and x = 1. o Any correct graph will have local minima at x = 1 and x = 3. 4. a. all real numbers except x = 1 b. x = 1.5 c. Y1 = f( and Y2 = x + 3. Graph and find intersections. Answers: x 2.449 and x 2.449 d. Use either method (switch x/y or composition). Answer: f 1 ( = 5. a. Outputs: 16, 9, 4, 4, 4, 4, 4, 9, 16. b. see graph at the right c. even symmetry d. No, because the graph does not pass the Horizontal Line Test (for example, (3, 9) and ( 3, 9) are both on the graph). e. For any point (except on the y-axis) of an even function s graph, there will be a reflected image across the y-axis, and such a pair of points will cause the function to fail the Horizontal Line Test, therefore not having an inverse. x 3 x 2 6. a. b. x (f g)( x ( f g)( 15 50 15 10 10 50 10 15 5 0 5 10 0 50 0 15 5 50 5 10 10 150 10 15 15 50 15 10 c. ( f / g)( and ( g f )(