Calculus Unit 1, Lesson 2: Composite Functions DATE: Objectives The students will be able to: - Evaluate composite functions using all representations Simplify composite functions Materials and Handouts - Warm-Up - Answers to homework #1 - Keynote and notes template - Tic Tac Toe grids - Homework #2 Homework #1-2 1) Composite Functions 2) Practice Skills Test (#1-1) Time Activity 20 min Homework Check / Warm-up - Warm up - Students check their answers to the homework and correct possible mistakes - Students are working on the warm-up: o Find domain and range of a graph o Evaluate various inputs for the graph, and an equation 25 min Lecture / Activity - Warm-up - Main concept: Composite Functions: f(g(x)) - Arrow maps of f and g. Evaluate various inputs into f that require operations to be performed first.. - Composition problem. Give groups time to see if they can figure out how to evaluate the problem before explaining. Repeat with other examples, including one that can t be determined, and one where the outside function is g. - Repeat the process, using a graph. - Repeat the process, using equations. - Independent Practice: Give several independent practice problems in equation form. - Show the functions f(x) = -3x + 4 and g(x) = 1 5x. Ask students to evaluate: f(-5); f(g(3)); f(δ); f(x + 2); f(g(x)). Each time, show how the input is substituted in for the x- value in f(x). In the last example, substitute in g(x) to show that f(g(x)) = -3(g(x)) + 4 before simplifying. - Show the functions f(x) = x 2 2x + 3 and g(x) = 4x 2. Simplify both f(g(x)) and g(f(x)). Point out that you get a different result depending on the order; composition is not commutative. - Show what happens when one of the functions is a constant. 20 min Classwork Give each group a Relay Race checker. For each round, when all the members of the group have their sheets completed and correct, the group earns a stamp and moves on to the next round. At the end of the lesson, each member of the group gets one chocolate kiss for each stamp earned. 5 min Closure Students: o Write down notes template in their logs/planners o Write down practice sheet and answer key in their logs/planners o Write down homework in their logs/planners
DATE: Calculus Section: Name: Unit 1, Lesson 2: Warm-Up 1) Determine if each graph represents a function. 2) Find the domain and range of each graph.
DATE: Calculus Section: Name: Unit 1, Lesson 2: Lecture Notes Main Concept Evaluating Functions 8 0.5-12 6 1 4 3-2 10 0 f ( 10 2) = f ( 16) = 7 f 14 = Evaluating Composite Functions f 6-1 2 0-4 7 5 g x y -4 8 9-1 15 0 2 6 f ( g(9) ) = f ( g(15) ) = f ( g( 4) ) = g( f ( 1) ) = f ( g(2) ) = g( f ( 5) ) =
f ( g(2) ) = g( f ( 3) ) = You try: f ( g( 1) ) = f ( f ( 4) ) = f (x) = x 2 2x + 3 g(x) = 4x 2 Is f ( g(x) ) = g( f (x) )? Watch out for constants! f (x) = x 2 2x + 3 g(x) = 8
Group Members: Relay Race Progress Checker Round Completed! Round Completed! 1 4 2 5 3 6 Group Members: Relay Race Progress Checker Round Completed! Round Completed! 1 4 2 5 3 6
Round 1: f (x) = 2x + 3 g(x) = x 2 1 h(x) = 7 g( h(x) ) = h( g(x) ) = Round 1: f (x) = 2x + 3 g(x) = x 2 1 h(x) = 7 g( h(x) ) = h( g(x) ) =
Round 2: f (x) = x 2 3x 1 g(x) = 5x + 2 h(x) = 3x f ( h(x) ) = h( f (x) ) = Round 2: f (x) = x 2 3x 1 g(x) = 5x + 2 h(x) = 3x f ( h(x) ) = h( f (x) ) =
Round 3: f (x) = x 3 g(x) = 3 x h(x) = 3 4x 2 g( g(x) ) = h( f (x) ) = h( g(x) ) = g( h(x) ) = Round 3: f (x) = x 3 g(x) = 3 x h(x) = 3 4x 2 g( g(x) ) = h( f (x) ) = h( g(x) ) = g( h(x) ) =
Round 4: f (x) = 5 g(x) = 2x 2 x h(x) =10 3x g( f (x) ) = f ( g(x) ) = h( h( f ( x ) ) ) = g( h(x) ) = Round 4: f (x) = 5 g(x) = 2x 2 x h(x) =10 3x g( f (x) ) = f ( g(x) ) = h( h( f ( x ) ) ) = g( h(x) ) =
Round 5: f (x) = 1 4 x + 2 g(x) = x 2 + 1 2 x 1 h(x) = 3 4 f ( h(x) ) = h( f (x) ) = g( h(x) ) = f ( g(x) ) = Round 5: f (x) = 1 4 x + 2 g(x) = x 2 + 1 2 x 1 h(x) = 3 4 f ( h(x) ) = h( f (x) ) = g( h(x) ) = f ( g(x) ) =
Round 6: f (x) = 6 + 0.4x g(x) = 0.1x 2 3.5x + 2 h(x) = 0.9x h( g(x) ) = g( h(x) ) = Round 6: f (x) = 6 + 0.4x g(x) = 0.1x 2 3.5x + 2 h(x) = 0.9x h( g(x) ) = g( h(x) ) =
DATE: Calculus Section: Name: Homework #1-2 Composite Functions f (x) = 4 2x g(x) = 2x 2 3x + 5 h(x) = 3x 2 x 10 1. f ( h(6) ) = 2. h( g(0) ) = 3. f ( f ( 5) ) = 4. g( f (1) ) = 5. f ( h( g(3) ) ) = You have a skills test next class. Make sure to prepare! 1) #1-1 part 1 (Do the attached problems. Be ready!)
Determine if it s a function (graphically) 1. Is each relation a function? Write yes or no below each one. y y y x x x y x Determine if it s a function (tables, arrow maps) 2. Is each relation a function? Write yes or no below each one. a) b) 5 1-2 0 3 2 ½ x 8 3 4-2 8 7 y 6 3 0-5 -1 8 Determine domain and range (tables, arrow maps) 3. Write the domain and range for the relations in problem 2. a) Domain: b) Domain: Range: Range: 4. Write the domain and range for the graph of f(x) below. Determine domain and range (graphically) Domain: Range: f(x)
5. Given that f(x) = 2x 2 + 5x 4, find f(-3). Evaluate functions (algebraically) 6. Use the graph of f(x) to evaluate each: a) f(-4) = Evaluate functions (graphically) b) f(4) = c) f(-51) = d) f(0) = 7. Use the table of values to evaluate each: Evaluate functions (with tables) f g a) f(0) = b) g(1) = x 5-5 3 8 0 1 9 y 12 10-1 3-5 0 7 x -3 2 8 1-4 7 6 y 0-5 -5 8 1 3 5