1 Wyner PreCalculus Fall 2013

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1 1 Wyner PreCalculus Fall 2013 CHAPTER ONE: FUNCTIONS AND THEIR GRAPHS Summary, Terms, and Objectives Most of calculus and precalculus is based on functions. A function is a process that takes one or more inputs and each time results in a specific output. For example, the function Birthday takes the input Johnny Depp and yields an output of June 9. When functions are algebraic, such as f(x) = x + they can be added, subtracted, multiplied, divided, combined through a process called composition, and undone by their inverse. They also can be graphed, by letting the input be x and the output be y. These graphs can be moved, stretched, and flipped, and each such transformation is based on a specific algebraic adjustment to the original function. The majority of this chapter is from Algebra II. 1-A Functions Thursday, 8/29 relation independent variable dependent variable function vertical line test argument ➊ Determine whether or not a relation is a function. ➋ Identify the argument of a function. ➌ Use function notation. 1-B Domain and Range Tuesday, 9/3 domain range ➊ Identify the domain and range of a graphed function. ➋ Identify the domain of a function in function notation. 1-C Composition and Inverses Thursday, 9/ composition inverse one-to-one horizontal line test ➊ Evaluate compositions of functions. ➋ Find an expression for a composition of functions. ➌ Identify the inverse of a basic function logically. ➍ Identify the inverse of a relation conceptually. ➎ Determine whether or not two functions f and g are inverses of each other. ➏ Find the inverse of a relation algebraically. ➐ Find the inverse of a relation graphically. ➑ Determine whether or not the inverse of a graph is a function. 1-D Graphs on Calculators Monday, 9/9 ➊ Graph a function on the calculator. ➋ Change the viewing area of the screen automatically. ➌ Change the viewing area of the screen manually to a specific region.

2 2 Wyner PreCalculus Fall E Transformation of Functions Thursday, 9/12 transformation pre-image image translation stretch reflection ➊ Translate a function h units right and k units up. ➋ Stretch a function horizontally by a factor of b and vertically by a factor of a. ➌ Reflect a function across the y-axis and/or the across the x-axis. ➍ Apply multiple transformations to a function. ➎ Given the graph of f(x), sketch f(x h) + k. ➏ Given the graph of f(x), sketch a f( x / b ). ➐ Given the graph of f(x), sketch f(-x) or -f(x). ➑ Use the equation of a pre-image to find the equation of a graph. Review Thursday, 9/12 Test Thursday, 9/19

3 3 Wyner PreCalculus Fall A Functions A RELATION is a relationship between two variables. An INDEPENDENT Variable (often x) is a variable that gets plugged into a relation. A DEPENDENT Variable (often y) is the variable equal to the value (result) of the relation. A FUNCTION is a relation in which every value of the independent variable yields no more than one value of the dependent variable. A graph that cannot be touched in more than one point at a time by any vertical line passes the VERTICAL LINE TEST and is a function. ➊ Determine whether or not a relation is a function. 1. It is not a function if there exists a value that could be plugged in that would result in multiple answers. ➊ Identify which of the following are functions. For those that are not, demonstrate this by giving a single value of x that results in two different values of y. a) The parabola at right is not a function because, for example, when x = 0, y is both 2 and -2. b) y = ± x is not a function because, for example, when x = 9, y is both 3 and -3. c) y = x is a function because the symbol is defined to represent only the nonnegative square root. d) The set of ordered pairs { (3, ), (, 2), (3, 9) } is not a function only because when x = 3, y is both and 9. e) y = the team that won the Superbowl in year x is a function because each Superbowl year had exactly one winner. - 0 f) y = the year team x won the Superbowl is not a function because, for example, when x = Green Bay, y is 1967, 1968, 1997, and The ARGUMENT of a Function is the expression plugged into the function. Arguments are placed within parentheses when it is otherwise not clear what is and is not part of the argument. ➋ Identify the argument of a function. 1. Identify the expression that is actually being affected by that specific function. ➋ For the equation 2x 1 12 = 4 cos (x 20) + tan 6x 20, identify the argument of the following functions. a) the square root function b) the cosine function c) the tangent function 2x 1 x 20 6x Function Notation uses f(x) instead of y, where f is the name of the function, x is the argument of the function, and f(x) is the resulting value (the dependent variable). Other letters are often used instead of f and x. ➌ Use function notation. 1. Use f(x) as the dependent variable. 2. If the independent variable is not x, change x to this variable. 3. You can change the name of the function from f to another letter to make it more clear what it is for or to distinguish between different functions. ➌ Write the following in function notation, and then plug in for each independent variable. a) y = x 2 + 3x b) A = πr 2 c) Mark makes $10 an hour. f(x) = x 2 + 3x A(r) = πr 2 M(h) = 10h f() = 40 A() = 2π M() = 0 -

4 4 Wyner PreCalculus Fall B Domain and Range 10 The DOMAIN of a Function is the set of all possible values of independent variable. 8 6 The RANGE of a Function is the set of all possible values of the dependent variable. 4 ➊ Identify the domain and range of a graphed function The domain is all x values the graph reaches The range is all y values the graph reaches. -4 ➊ Identify the domain and range of the function graphed at right The domain is approximately -10 x The range is approximately - y 6. In this class, domain and range only apply to real numbers, and numbers that result in a nonreal value, such as x = -1 for f(x) = x, are not considered to be in the domain of the function. The domain of a fraction is all values of x that result in the denominator being nonzero. The domain of a square root function or other even root function is all values of x that result in the argument being nonnegative. The domain of a logarithmic function is all values of x that result in the argument being positive. ➋ Identify the domain of a function in function notation. 1. If there is a fraction, solve A 0 where A is the denominator. 2. If there is an even root, solve A 0 where A is the argument of the root. 3. If there is a logarithm, solve A > 0, where A is the argument of the logarithm. 4. Combine the limitations from steps 1-3. The domain is all real numbers other than these. ➋ Identify the domain of the following functions. 1 a) a(x) = x b) b(x) = x c) c(x) = x d) d(x) = log x all real numbers x 0 x 0 x > 0 1 2x 8 e) e(x) = 2x 8 f) f(x) = g) g(x) = 2x 8 h) h (x) = log (2x 8) i) i(x) = log (2x + 7) 3 20 x 1. 2x x x 8 > x 0 all real numbers x 4 x 4 x > 4 x x 0 x x + 7 > 0 x > -7 / / 2 < x 20, x

5 Wyner PreCalculus Fall C Composition and Inverses COMPOSITION of Functions is plugging an entire function, or its value, into a function. The composition of the function f with the function g is written f(g(x)) or (f o g)(x). ➊ Evaluate compositions of functions. 1. Calculate the value of the inner or last function. 2. Plug this value into the outer or first function. ➊ Given f(x) = 4x 10 and g(x) = x 2 + 2x 3, evaluate the following. a) f(g(3)) b) g(f(3)) 1. f( ) g(4 3 10) f(12) g(2) = (2) 3 = ➋ Find an expression for a composition of functions. 1. Substitute the inner or last function s expression, in parentheses, for each variable in the outer or first function. 2. Simplify. ➋ Using the functions f and g, above, give an expression for the following. a) g(f(x)) b) f(g(x)) 1. g(4x 10) = (4x 10) 2 + 2(4x 10) 3 f(x 2 + 2x 3) = 4(x 2 + 2x 3) x 2 40x 40x x x 2 + 8x x 2 72x x 2 + 8x 22 The INVERSE of a Function is the function that undoes the original function. If the inverse is also a function, then f is ONE-TO-ONE and the inverse function is labeled f -1. f -1 (f(x)) = x ➌ Identify the inverse of a basic function by definition. The inverse of addition is subtraction. The inverse of multiplication is division. The inverse of a power is a root. The inverse of an exponential is a logarithm. ➌ Identify the inverse of each of the following functions, and verify that f -1 (f(x)) = x. a) a(x) = x + a -1 (x) = x a -1 (a(x)) = x + = x b) b(x) = x b -1 (x) = x / b -1 (b(x)) = x / = x c) c(x) = x c -1 (x) = x 1/ c -1 (c(x)) = (x ) 1/ = x d) d(x) = x d -1 (x) = log x d -1 (d(x)) = log x = x ➍ Identify the inverse of a function conceptually. Switch the variables so that the independent variable becomes the dependent variable and vice versa. ➍ f(x) = x /7 is the number of miles Sean can run in x minutes. (Minutes are plugged in to calculate miles.) f -1 (x) = 7x is the number of minutes it takes Sean to run x miles. (Miles are plugged in to calculate minutes.)

6 6 Wyner PreCalculus Fall 2013 ➎ Determine whether or not two functions f and g are inverses of each other. 1. Find f(g(x)) or g(f(x)). 2. If it equals x, the functions are inverses of each other. ➎ Are f(x) = 4x + 8 and g(x) = x /4 8 inverses of each other? 1. f(g(x)) = 4( x /4 8) + 8 = x = x x 24 x, so f and g are not inverses of each other. The inverse of a relation can be found by switching the independent variable with the dependent variable. ➏ Find the inverse of a relation algebraically. 1. Switch the independent variable with the dependent variable. y can be written instead of f(x). 2. Solve for the new dependent variable. 3. If the new equation is a function, write the dependent variable as f -1 (x). ➏ Find the inverse of the following functions. a) f(x) = x 8 b) g(x) = 2x + 1 c) h(x) = x 2 1. x = y 8 x = 2y + 1 x = y 2 2. y = x + 8 y = x / 2 1 / 2 y = ± x 3. f -1 (x) = x + 8 g -1 (x) = x / 2 1 / 2 The graph of a relation s inverse can be seen by reflecting the graph of the original relation across the y = x diagonal. ➐ Find the inverse of a relation graphically. 1. Draw the line y = x. 2. Reflect the graph of the original relation across this line. ➐ y = x The inverse of a vertical line is a horizontal line. Therefore, since the vertical line test identifies whether or not a graph is a function, the HORIZONTAL LINE TEST identifies whether or not the inverse of a graph is a function. ➑ Determine whether or not the inverse of a graph is a function. 1. The inverse is a function unless there exists a horizontal line that touches the graph at more that one point. ➑ The inverse of the function graphed at right is not a function because, for example, the horizontal line y = 0-0 passes through it at x = -2 and at x = 2. -

7 7 Wyner PreCalculus Fall D Graphs on Calculators The calculator can graph functions. ➊ Graph a function on the calculator. 1. Solve the equation for y. 2. Type the function, starting with the [Y=] button. 3. Push [GRAPH]. ➋ Change the viewing area of the screen automatically. 1. Push [ZOOM]. 2. Choose one of the zoom options. For example: ZStandard is the calculator s default screen, showing -10 x 10 and -10 y 10. ZFit attempts to guess the best screen fit for your graph. Zoom In or Zoom Out zooms in or out each time you push [ENTER] after you select it. ➌ Change the viewing area of the screen manually to a specific region. 1. Push [WINDOW]. 2. Type the set of x values you want the graph to show by changing Xmin and Xmax. 3. Repeat step 2 for y. ➌ Set the viewing area to -3 x 3 and - y Xmin=-3 Xmax=3 3. Ymin=- Ymax=40

8 8 Wyner PreCalculus Fall E Transformations of Functions A TRANSFORMATION takes an original function, called the PRE-IMAGE, and changes its graph s position, size, or direction so that it becomes a new function, called the IMAGE. Three main types of transformations are translations, stretches (or dilations), and reflections. A TRANSLATION moves the graph of a function. f(x h) is a Horizontal Translation: The graph is moved h units to the right. f(x) + k is a Vertical Translation: The graph is moved k units upward. ➊ Translate a function h units right and k units up. 1. Subtract h from each x. To translate it left, h will be negative, which will result in adding to x. Add k to the expression. To translate it down, k will be negative, which will result in subtracting from the expression. 2. Simplify. ➊ Translate the pre-image f(x) = 2x 2 x 4 three units left and five units up. 1. f(x + 3) + = 2(x + 3) 2 (x + 3) f(x + 3) + = (2x x + 18) (x + 1) + 1 f(x + 3) + = 2x 2 + 7x + 4 A STRETCH increases or decreases the size of the graph by expanding it away from or compressing it toward the axis. f( x / b ) is a Horizontal Stretch: The graph is expanded away from the y-axis by a factor of b. a f(x) is a Vertical Stretch: The graph is expanded away from the x-axis by a factor of a. ➋ Stretch a function horizontally by a factor of b and vertically by a factor of a. 1. Divide each x by b. To compress the function, b will be less than 1, which will result in multiplying. Multiply the expression by a. To compress the function, a will be less than 1, which will result in dividing. 2. Simplify. - 0 ➋ Stretch the pre-image f(x) = 2 + sin x by a factor of 2 vertically (twice as tall) and by a factor of 1 / 3 horizontally (one third as wide) f( x / 1/3 ) = 2(2 + sin x / 1/3 ) 2. 2 f(3x) = sin 3x

9 9 Wyner PreCalculus Fall 2013 A REFLECTION flips the graph across a line. f(-x) is a Horizontal Reflection: The graph is flipped across the y-axis. -f(x) is a Vertical Reflection: The graph is flipped across the x-axis. ➌ Reflect a function across the y-axis and/or the across the x-axis. 1. To reflect horizontally (across the y-axis), multiply each x by -1. To reflect vertically (across the x-axis), multiply the expression by Simplify. ➌ Reflect the pre-image f(x) = x x + 3 across the y-axis and across the x-axis. 1. -f(-x) = -((-x) 2 + 4(-x) + 3) 2. -f(-x) = -(x 2-4x + 3) -f(-x) = -x 2 + 4x 3 When different types of transformations are each applied to a function, the end result will depend on the order in which the transformations occur. ➍ Apply multiple transformations to a function. 1. Do each transformation in the order stated. See ➊, ➋, and ➌. 2. Simplify. ➍ Transform the pre-image f(x) = x 2 6x + 9 by doing the following. a) Reflect it across the x-axis and then translate it up four units. b) Translate it up four units and then reflect it across the x-axis. 1. -f(x) = -(x 2 6x + 9) f(x) + 4 = x 2 6x f(x) + 4 = -(x 2 6x + 9) + 4 -(f(x) + 4) = -(x 2 6x ) 2. -f(x) + 4 = -x 2 + 6x -f(x) 4 = -x 2 + 6x

10 10 Wyner PreCalculus Fall 2013 Graphs can be transformed based on the rules above. ➎ Given the graph of f(x), sketch f(x h) + k. 1. Identify h and k. 2. Translate the graph h units to the right and k units up. ➎ Given f(x) in blue, graph f(x + 3) h = -3, k = 1 ➏ Given the graph of f(x), sketch a f( x / b ). 1. Identify a and b. 2. Multiply every y coordinate by a. Every point will be a times as far from the x axis as it was before. 3. Multiply every x coordinate by b. Every point will be b times as far from the y axis as it was before. ➏ Given f(x) in blue, graph 2f(2x). 1. a = 2, b = 1 / 2 ➐ Given the graph of f(x), sketch f(-x) or -f(x). 1. f(-x) is a horizontal reflection, so reflect the graph across the y-axis. Every positive x-coordinate will become negative and every negative x-coordinate becomes positive. 2. -f(x) is a vertical reflection, so reflect the graph across the x-axis. Every positive y-coordinate will become negative and every negative y-coordinate becomes positive. ➐ Given f(x) in blue, graph f(-x). and -f(x). The equation of an image can be determined by identifying the transformations that have been applied to the pre-image. ➑ Use the equation of a pre-image to find the equation of a graph. 1. Identify the equation of the pre-image. 2. If there is a translation see ➊. 3. If there is a stretch see ➋. 4. If there is a reflection see ➌.. Simplify. ➑ Write the equation for g(x) graphed at right. 1. The pre-image is f(x) = x 2 6x There is a vertical translation up : f(x) + = x 2 6x There is a horizontal translation left 4: f(x + 4) + = (x + 4) 2 6(x + 4) g(x) 4. There is a vertical reflection: -(f(x + 4) + ) = -((x + 4) 2 6(x + 4) ). g(x) = -(f(x + 4) + ) = -x 2 2x 1 f(x) = x 2 6x + 4 f(x + 3) + 1 f(-x) f(2x) f(x) f(x) f(x) -f(x)