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9/5/2006Pre-Calculus R R { [ 4, ) } { (- , 3 ] } { R \ { 2 } } { R \ { 1 } } { R \ { -3, 0 } } R { (- 3, ) } { (- , 4 ] U [ 2, ) } { (- , -1) U [ 0, ) } { [ 0, ) } R { [ -8, ) } { [ 0, ) } { R \ { ½ } }
![9/5/2006Pre-Calculus R R { [ 4, ) } { (- , 3 ] } { R \ { 2 } } { R \ { 1 } } { R \ { -3, 0 } } R { (- 3, ) } { (- , 4 ] U [ 2, ) } { (- , -1) U [ 0, ) } { [ 0, ) } R { [ -8, ) } { [ 0, ) } { R \ { ½ } }](http://images.slideplayer.com./20/6212944/slides/slide_1.jpg "9/5/2006Pre-Calculus R R { [ 4, ) } { (- , 3 ] } { R \ { 2 } } { R \ { 1 } } { R \ { -3, 0 } } R { (- 3, ) } { (- , 4 ] U [ 2, ) } { (- , -1) U [ 0, ) } { [ 0, ) } R { [ -8, ) } { [ 0, ) } { R \ { ½ } }")
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9/5/2006Pre-Calculus
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9/5/2006Pre-Calculus continuous discontinuous infinite discontinuous removable continuous discontinuous removable discontinuous jump discontinuous - jump continuous discontinuous - infinite continuous discontinuous - infinite
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9/5/2006Pre-Calculus
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9/5/2006Pre-Calculus (3x+4)(x 1) jump (x^3+1)(x 0)+ (2)(x=0) removable (3+x 2 )(x -2) (x 1) jump
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9/5/2006Pre-Calculus incr: (- , ) decr: (- , 0 ] incr: [ 0, ) decr: (- , 0 ] incr: [ 0, ) decr: [ - 1, 1 ] incr: (- , -1 ], [ 1, ) decr: [ 3, 5 ], incr: [ , 3 ] constant: [ 5, ) decr: [ 3, ), incr: ( 0 ] constant: [ 0, 3) decr: ( - , ) decr: (- , -8 ] incr: [ 8, ) decr: ( - , 0 ] incr: [ 0, 3 ) constant: [ 3, ) decr: ( 0, ) incr: ( - , 0 ) decr: ( 2, ) incr: ( - , 2) constant: [ -2, 2 ] decr: ( - , 7 ) decr: ( 7, )
![9/5/2006Pre-Calculus incr: (- , ) decr: (- , 0 ] incr: [ 0, ) decr: (- , 0 ] incr: [ 0, ) decr: [ - 1, 1 ] incr: (- , -1 ], [ 1, ) decr: [ 3, 5 ], incr: [ , 3 ] constant: [ 5, ) decr: [ 3, ), incr: ( 0 ] constant: [ 0, 3) decr: ( - , ) decr: (- , -8 ] incr: [ 8, ) decr: ( - , 0 ] incr: [ 0, 3 ) constant: [ 3, ) decr: ( 0, ) incr: ( - , 0 ) decr: ( 2, ) incr: ( - , 2) constant: [ -2, 2 ] decr: ( - , 7 ) decr: ( 7, )](http://images.slideplayer.com./20/6212944/slides/slide_7.jpg "9/5/2006Pre-Calculus incr: (- , ) decr: (- , 0 ] incr: [ 0, ) decr: (- , 0 ] incr: [ 0, ) decr: [ - 1, 1 ] incr: (- , -1 ], [ 1, ) decr: [ 3, 5 ], incr: [ , 3 ] constant: [ 5, ) decr: [ 3, ), incr: ( 0 ] constant: [ 0, 3) decr: ( - , ) decr: (- , -8 ] incr: [ 8, ) decr: ( - , 0 ] incr: [ 0, 3 ) constant: [ 3, ) decr: ( 0, ) incr: ( - , 0 ) decr: ( 2, ) incr: ( - , 2) constant: [ -2, 2 ] decr: ( - , 7 ) decr: ( 7, )")
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9/5/2006Pre-Calculus
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9/5/2006Pre-Calculus unbounded bounded below b = 0 bounded below b = 1 unbounded bounded above B = 0 bounded b= -1, B = 1 bounded below b = 0 bounded below b = -1 bounded below b = 0 bounded above B = 0 Right branch: bounded below b = 5 Left branch: bounded above B = 5
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9/5/2006Pre-Calculus y-axis EVEN functions The graph looks the same to the left of the y-axis as it does to the right For all x in the domain of f, f(-x) = f(x) x-axis The graph looks the same above the x-axis as it does below it (x, - y) is on the graph whenever (x, y) is on the graph origin ODD functions The graph looks the same upside Down as it does right side up For all x in the domain of f, f(-x) = - f(x)
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9/5/2006Pre-Calculus Algebraic Test for even/odd/neither: Replace all x with –x:
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9/5/2006Pre-Calculus Algebraic Test for even/odd/neither: Replace all x with –x: f(x)=f(-x) so even! Now try g(-x):
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9/5/2006Pre-Calculus Algebraic Test for even/odd/neither: Replace all x with –x: f(x)=f(-x) so even! g(-x)=-g(x) so odd!
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9/5/2006Pre-Calculus Algebraic Test for even/odd/neither: Replace all x with –x:
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9/5/2006Pre-Calculus Algebraic Test for even/odd/neither: Replace all x with –x: This is neither the same nor opposite so it is neither!
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9/5/2006Pre-Calculus Odd Even Odd Even Neither Even Neither Even Odd
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9/5/2006Pre-Calculus Students will be able to determine the vertical and Horizontal asymptotes of functions by inspecting their graphs Do Now: use your graphing calculator to graph:
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9/5/2006Pre-Calculus vertically horizontally
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9/5/2006Pre-Calculus horizontally vertically will not actually touch asymptotes tan and cot x = -1 x = 2 y = 0 End behavior Limit notation
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9/5/2006Pre-Calculus Students will be able to determine the vertical and Horizontal asymptotes of functions by inspecting their graphs Graph the function above and determine the vertical And the horizontal asymptotes.
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9/5/2006Pre-Calculus Students will be able to determine the vertical and Horizontal asymptotes of functions by inspecting their graphs The end behavior of the function above is related to the Horizontal asymptote. Vertical: Horizontal:
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9/5/2006Pre-Calculus We read this as “the limit as x approaches infinity is” Which means “as we look to the far right of the graph, What y value does it head near”
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9/5/2006Pre-Calculus Vertical: x = - 3 Horizontal: y = 0 Vertical: x = 2, -2 Horizontal: y = 0 Vertical: x = 3
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9/5/2006Pre-Calculus Student will be able to use function properties to Analyze a Function.
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9/5/2006Pre-Calculus Yes { ( - , -1 ) U (-1, 1) U (1, ) } Infinite discontinuities Decreasing: (- , -1), (-1, 0 ] Unbounded Left piece: B = 0, Middle piece b = 3, Right piece B = 0 Local min at (0, 3) Even Horizontal: y = 0, Vertical: x = -1, 1 Each x-value has only 1 y-value { ( - , 0) U [ 3, ) } Increasing: ([ 0, 1), (1, )
![9/5/2006Pre-Calculus Yes { ( - , -1 ) U (-1, 1) U (1, ) } Infinite discontinuities Decreasing: (- , -1), (-1, 0 ] Unbounded Left piece: B = 0, Middle piece b = 3, Right piece B = 0 Local min at (0, 3) Even Horizontal: y = 0, Vertical: x = -1, 1 Each x-value has only 1 y-value { ( - , 0) U [ 3, ) } Increasing: ([ 0, 1), (1, )](http://images.slideplayer.com./20/6212944/slides/slide_26.jpg "9/5/2006Pre-Calculus Yes { ( - , -1 ) U (-1, 1) U (1, ) } Infinite discontinuities Decreasing: (- , -1), (-1, 0 ] Unbounded Left piece: B = 0, Middle piece b = 3, Right piece B = 0 Local min at (0, 3) Even Horizontal: y = 0, Vertical: x = -1, 1 Each x-value has only 1 y-value { ( - , 0) U [ 3, ) } Increasing: ([ 0, 1), (1, )")
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9/5/2006Pre-Calculus What we just did was to “analyze a function” The sheet provided to you lists the aspects of A function to include in an analysis.
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9/5/2006Pre-Calculus In-class Exercise Section 1.3 In-class Exercise Section 1.3 Domain Range Continuity Increasing Decreasing Boundedness Extrema Symmetry Asymptotes End Behavior
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9/5/2006Pre-Calculus
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9/5/2006Pre-Calculus Yes { ( - , ) } continuous Decreasing: (- , 0 ] Bounded below b = 0 Absolute min = (0,0) Neither even or odd none Each x-value has only 1 y-value { [ 0, ) } Increasing: [ 0, ) { ( - , -3 ] U [ 7, ) }
![9/5/2006Pre-Calculus Yes { ( - , ) } continuous Decreasing: (- , 0 ] Bounded below b = 0 Absolute min = (0,0) Neither even or odd none Each x-value has only 1 y-value { [ 0, ) } Increasing: [ 0, ) { ( - , -3 ] U [ 7, ) }](http://images.slideplayer.com./20/6212944/slides/slide_30.jpg "9/5/2006Pre-Calculus Yes { ( - , ) } continuous Decreasing: (- , 0 ] Bounded below b = 0 Absolute min = (0,0) Neither even or odd none Each x-value has only 1 y-value { [ 0, ) } Increasing: [ 0, ) { ( - , -3 ] U [ 7, ) }")
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9/5/2006Pre-Calculus 10 Basic Functions
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9/5/2006Pre-Calculus Do Now: Add: Subtract: What are the domains of :
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9/5/2006Pre-Calculus
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9/5/2006Pre-Calculus f(x) + g(x) f(x) – g(x) f(x)g(x) f(x)/g(x), provided g(x) 0 3x 3 + x 2 + 6 3x 3 – x 2 + 8 3x 5 – 3x 3 + 7x 2 – 7 x 2 – (x + 4) = x 2 – x – 4
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9/5/2006Pre-Calculus Restricted Domains: Find the domains of the following functions:
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9/5/2006Pre-Calculus Restricted Domains: Find the domains of the following functions: 1.Find: and include the overlapping domain 2.Find and include the domain
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9/5/2006Pre-Calculus Restricted Domains: Find the domains of the following functions: Domain:
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9/5/2006Pre-Calculus Do Now: Simplify this “complex” fraction Student will be able to compose two functions And find their domains
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9/5/2006Pre-Calculus x2x2
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9/5/2006Pre-Calculus x2x2 sinx +,-,x,/ The squaring function the sine function composition f ○ g f(g(x) x2x2
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9/5/2006Pre-Calculus 1.Write the first function 2.Replace “x” with ( ) 3.Place the 2nd function in ( ) 4.Decide on the domain and simplify Steps for composition:
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9/5/2006Pre-Calculus x2x2 sinx +,-,x,/ The squaring function the sine function composition f ○ g f(g(x) x2x2 4x 2 – 12x + 9 2x 2 – 3 x4x4 1 5 4x-9
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9/5/2006Pre-Calculus The domains of compositions We need to “inherit” the domain of the second function And consider the new function as well! Use the graphing calculator to verify!
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9/5/2006Pre-Calculus Domain is all reals except -2
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9/5/2006Pre-Calculus For part b: (see “vars” which is next to clear for the y)
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9/5/2006Pre-Calculus Domain is all reals except -2 Domain is all reals exc. 0 and -.5
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9/5/2006Pre-Calculus Domain is all reals except -2 Domain is all reals exc. 0 and -.5 D: all reals except 0 D: all reals except -2 and -4
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9/5/2006Pre-Calculus Decomposition: Students will be able to decompose a composite function. Do Now: perform the composition:
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9/5/2006Pre-Calculus Given the function, find what f(x) and g(x) could be. Recall the steps for composition: 1.Write the first function 2.Replace “x” with ( ) 3.Place the 2nd function inside ( ) 4.Decide on the domain and simplify Ask yourself: what could be in the ( ) ?
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9/5/2006Pre-Calculus inside function outside function x 2 + 1 x2x2
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9/5/2006Pre-Calculus Practice examples:
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9/5/2006Pre-Calculus inside function outside function x 2 + 1 x2x2
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9/5/2006Pre-Calculus Hand in this example on an “exit card”
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9/5/2006Pre-Calculus inverse functions horizontal line test original relation Graph is a function (passes vertical line test. Inverse is also a function (passes horizontal line test.) both vertical and horizontal line test like A one-to-one function is paired with a unique y inverse function is paired with a unique x f –1 f –1 (b) = a, iff f(a) = b Graph is a function (passes vertical line test. Inverse is not a function (fails horizontal line test.)
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9/5/2006Pre-Calculus Do Now: State the domain and range of the following function: Student will be able to find the inverse function and state The inherited domain and range
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9/5/2006Pre-Calculus Domain Range: Get y terms together! Factor y out Note the “inherited“ domain
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9/5/2006Pre-Calculus D: { ( - , ) } R: { ( - , ) } D: { [ 0, ) } R: { [ 0, ) } D: { ( - , - 2) U ( -2, ) } R: { ( - , 1) U (1, ) } D: { ( - , ) } D: { [ 0, ) } D: { ( - , 1) U (1, ) }
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9/5/2006Pre-Calculus Functions that are not one to one require restrictions On their domains in order to make them 1 – 1 and Find their inverses! What is the domain and range?
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9/5/2006Pre-Calculus Functions that are not one to one require restrictions On their domains in order to make them 1 – 1 and Find their inverses! Domain: restricted to Range:
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9/5/2006Pre-Calculus Functions that are not one to one require restrictions On their domains in order to make them 1 – 1 and Find their inverses! Domain: Range:
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9/5/2006Pre-Calculus { ( - , ) } f(x) and g(x) are inverses { ( - , ) } { ( - , - 5) U ( - 5, ) } { ( - , ) }
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9/5/2006Pre-Calculus Yes passes horizontal line test Yes D: { ( - , 0 ) U ( 0, ) } R: { ( - , 4 ) U ( 4, ) } D: { ( - , 4 ) U ( 4, ) }
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9/5/2006Pre-Calculus D: { ( - , - 2 ) U ( - 2, 1 ) U ( 1, ) } D: { ( - , 0 ) U ( 0, ) } D: { ( - , 2/3 ) U ( 2/3, 1 ) U ( 1, ) }
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9/5/2006Pre-Calculus STUDENTS WILL BE ABLE TO NAME THE TRANSFORMATIONS OF A FUNCTION THAT TOOK PLACE WHEN GIVEN AN EQUATION DO NOW: worksheet
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9/5/2006Pre-Calculus add or subtract a constant to the entire function f(x) + cup c units f(x) – c down c units add or subtract a constant to x within the function f(x – c) right c units f(x + c) left c units
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9/5/2006Pre-Calculus multiply c to the entire function Stretch if c > 1 Shrink if c < 1 multiply c to x within the function A reflection combined with a distortion complete any stretches, shrinks or reflections first complete any shifts (translations) Stretch by C Shrink by
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9/5/2006Pre-Calculus reflections negate the entire function y = – f(x) negate x within the function y = f(-x)
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9/5/2006Pre-Calculus 1. What transformations took place with The basic absolute value function? 2. Write the equation that results by taking the squaring F function and: A vertical stretch factor of 4, shift left 6
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9/5/2006Pre-Calculus 1. What transformations took place with The basic absolute value function? 2. Write the equation that results by taking the squaring function and: A vertical stretch factor of 4, shift left 6 Vertical stretch by 5, horiz. Shrink by ½, vertical shift up 3
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9/5/2006Pre-Calculus AnswersAnswers AnswersAnswers y = 1/x 4 y = x, y = x 3, y = 1/x, y = ln (x) y = sqrt(x) y = ln(x) y = 2sin(0.5x) Stretch by 8 Shrink ½ Shrink by 1/8 Stretch by 2
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