Rewriting Absolute Value Functions as Piece-wise Defined Functions
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1 Rewriting Absolute Value Functions as Piece-wise Defined Functions Consider the absolute value function f ( x) = 2x Sketch the graph of f(x) using the strategies learned in Algebra II finding the vertex and using the slope of each side. Now, write the graphed function as a two-piece, piece-wise defined function. Constraint of Equation of Each Piece Each Piece ì ï f (x) = í ï ï î Follow the strategy outlined below for the function f ( x) = 2x Set each expression inside of absolute value = 0 and solve for x. 2. Draw a number line and divide it into intervals using the values of x found in the previous step and write the entire equation of the function in each interval, replacing the absolute value bars with parenthesis. 3. Determine the sign of each expression in parenthesis by choosing a value from each interval and substituting in for x. Place the sign in front of the expressions on each interval. 4. Simplify the equations on each interval by distributing the signs determined in the previous step and combining like terms. Then, write the equation as a piece-wise defined function. Compare your piece-wise defined function that you wrote in step #4 above to the equation that you wrote from the graph at the beginning of this exercise. Are they the same? Should they be the same? Explain. Daily Lessons and Assessments for Pre-AP* Calculus, A Complete Course Page 98
2 Follow the strategy outlined below for the function g ( x) = x+ 4+ x Set each expression inside of absolute value = 0 and solve for x. 2. Draw a number line and divide it into intervals using the values of x found in the previous step and write the entire equation of the function in each interval, replacing the absolute value bars with parenthesis. 3. Determine the sign of each expression in parenthesis by choosing a value from each interval and substituting in for x. Place the sign in front of the expressions on each interval. 4. Simplify the equations on each interval by distributing the signs determined in the previous step and combining like terms. Then, write the equation as a piece-wise defined function. Now, graph the function g(x) on the grid below using the piece-wise defined function and check your graph in the graphing calculator. Then, identify the domain and range of g(x). Domain: Range: Daily Lessons and Assessments for Pre-AP* Calculus, A Complete Course Page 99
3 Using the strategy defined above, rewrite each of the following functions as piece-wise defined functions and then graph the function on the grids provided. 1. f ( x) = x+ 2+ 2x- 3 Domain: Range: 2. g ( x) = -2 x Domain: Range: Daily Lessons and Assessments for Pre-AP* Calculus, A Complete Course Page 100
4 3. Rewrite the function bars. h( x) = x+ 1+ x- 3 x- 5 as a piece-wise defined function without absolute value Daily Lessons and Assessments for Pre-AP* Calculus, A Complete Course Page 101
5 Symmetry in Graphs Graphical, Numerical, and Analytical Symmetries can be two main types reflective or rotational. Specifically, a graph can have three different types of reflective symmetry and one type of point symmetry. A graph that has reflective symmetry A graph that has rotational symmetry Consider the four graphed functions below. In the box to the left of each graph, label it as having either reflective or rotational symmetry. Each represents a graph that has y axis symmetry, x axis symmetry, symmetry, or y = x symmetry. Study the graphs and label each accordingly in the boxes above each graph. Daily Lessons and Assessments for Pre-AP* Calculus, A Complete Course Page 103
6 Origin Symmetry Graphically describe a graph that has origin rotational symmetry. Numerically describe a graph that has origin rotational symmetry. Analytically, how could you determine whether an equation has a graph that will exhibit origin rotational symmetry? Y Axis Symmetry Graphically describe a graph that has y axis reflective symmetry. Numerically describe a graph that has y axis reflective symmetry. Analytically, how could you determine whether an equation has a graph that will exhibit y axis reflective symmetry? Daily Lessons and Assessments for Pre-AP* Calculus, A Complete Course Page 104
7 Y = X Symmetry Graphically describe a graph that has y = x reflective symmetry. Numerically describe a graph that has y = x reflective symmetry. Analytically, how could you determine whether an equation has a graph that will exhibit y = x reflective symmetry? X Axis Symmetry Graphically describe a graph that has x axis reflective symmetry. Numerically describe a graph that has x axis reflective symmetry. Analytically, how could you determine whether an equation has a graph that will exhibit x axis reflective symmetry? Daily Lessons and Assessments for Pre-AP* Calculus, A Complete Course Page 105
8 Analytically determine the symmetry, if any exists, that the graphs of the following equations will exhibit. 1) x 2 + 2y 2 = 3 2) y 4 + 2x = 3x 3) 2x 2 3 y = 5 4) x 2 + y 2 = 25 Daily Lessons and Assessments for Pre-AP* Calculus, A Complete Course Page 106
9 Even and Odd Functions In geometry, you learned about two types of symmetry reflective and rotational symmetry. Determine which of the following graphs exhibit reflective symmetry and which graphs represent rotational symmetry. For those that exhibit reflective symmetry, draw the line of symmetry using a dashed line. For those that exhibit rotational symmetry, identify the point of symmetry. F(x) G(x) H(x) P(x) R(x) Q(x) Daily Lessons and Assessments for Pre-AP* Calculus, A Complete Course Page 109
10 On the previous page, there are three functions that exhibit reflective symmetry and their line of symmetry is the y axis. Also, there are three functions that exhibit rotational symmetry and their point of symmetry is the origin. In the boxes below, identify the three functions that have reflective symmetry and the three functions that have rotational symmetry. Then, in the boxes below each function, list all of the specifically plotted ordered pairs on the graph. List them in order that they appear from left to right on the graph. Functions with Y Axis Reflective Symmetry Functions with Origin Rotational Symmetry Daily Lessons and Assessments for Pre-AP* Calculus, A Complete Course Page 110
11 What observation do you make about all of the points on the graphs of the functions with y axis reflective symmetry? What observation do you make about all of the points on the graphs of the functions with origin rotational symmetry? Numerical and Graphical Definition of Even and Odd Functions Even Function Odd Function Daily Lessons and Assessments for Pre-AP* Calculus, A Complete Course Page 111
12 Analytically, a function is an even function if the equation of the function f( x) can be simplified to be the same equation as that of the function f(x). Thus,,f( x) = f(x). Example: f(x) = 23x + x 2 Example: 4 g ( x) = 2x + x 2 Analytically, a function is an odd function if the equation of the function f( x) can be simplified to be the OPPOSITE of the function f(x). Thus, f( x) = f(x). Example: 3 = 3x Example: g( x) = 2x- x 2x f ( x) + x If f( x) does not simplify to the same equation as f(x) or f(x), then the function is neither even nor odd. Example f(x) = 3x 2 + x Example: g(x) = 2x x + x 2 Daily Lessons and Assessments for Pre-AP* Calculus, A Complete Course Page 112
13 One to One Functions Numerical, Graphical, and Analytical Approaches A function is If the point (2, 3) is on the graph of a function, f(x), then no other point can have a(n) of. However, it is possible for another point on the graph of f(x) to have a y coordinate of 3. For example, both (2, 3) and (5, 3). A one-to-one function is For example, if (2, 3) and (5, 3) are both on the graph of f(x), then there is a y value, namely 3, that has more than one x value. Therefore, there is no way that the function is a one to one function. Numerical Determination if a function is a one-toone function. Graphical Determination if a function is a one-toone function. Analytical Determination if a function is a one-toone function. The inverse of a function is defined to be What property of the function must exist in order for the inverse of the function to exist? Why? Daily Lessons and Assessments for Pre-AP* Calculus, A Complete Course Page 117
14 If it can be determined that a function is one-to-one, then the inverse of the function can be found. Fill in the following chart which will describe how to find the inverse of the function if it is established to be a one-to-one function. Numerical Determination of the inverse of a function Graphical Determination of the inverse of a function Analytical Determination of the inverse of a function Examples: For each of the following functions, determine if the inverse of the function exists, providing justification. If it does, then find the inverse. If the relation is given in numerical form, give the table of values that would represent f 1 (x). If the relation is given in graphical form, sketch the graph of f 1 (x). If the relation is given in analytical form, find the equation of f 1 (x). a. b. c. Graph of f(x) d. Graph of g(x) Daily Lessons and Assessments for Pre-AP* Calculus, A Complete Course Page 118
15 e. Graph of h(x) f. f(x) = (x 3) 2 2 g. f ( x) = 2 x- 3 h. g ( x) = 2 x Daily Lessons and Assessments for Pre-AP* Calculus, A Complete Course Page 119
16 3 4 Let g ( x) = x-. 2 Find g -1 ( x ) Find g(g 1 (x)). Find g 1 (g(x)). If two functions, f(x) and g(x), are inverses of each other, then there are two special composite functions that will equal the same thing: Examples: Determine if each of the pairs of functions below are inverse functions of each other or not. Show your work. a. f(x) = 2x + 3 g(x) = x- 3 2 x b. f(x) = g(x) = 2x + 1 Daily Lessons and Assessments for Pre-AP* Calculus, A Complete Course Page 120
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