Ppt on domain and range of functions

Hi, my name is RaD My radar detects Range and Domain Watch my radar (yellow line) passing through the graph in the next slide.

is RaD My radar detects Range and Domain Watch my radar (yellow line) passing through the graph in the next slide The DOMAIN is the set of all possible x values for that function The DOMAIN is the set of all possible x values for that function What values can be substituted for x? Example The RANGE is the set of all possible y values for that function What values are possible/


8.2 The Reciprocal Function Family Honors. The Reciprocal Functions The Reciprocal function f(x) = x ≠0 D: {x|x ≠ 0} R: {y|y ≠ 0} Va: x = 0 Ha: y = 0.

are hyperbolas Goal: Graph simple rational functions Hyperbola A type of rational function. Has 1 vertical asymptote and 1 horizontal asymptote. Has 2 parts called branches. (blue parts) They are symmetrical. We’ll discuss 2 different forms. x=0 y=0 Hyperbola (continued) One form: Has 2 asymptotes: x = h (vert.) y = k (horiz.) Ex: Graph State the domain & range. Vertical Asymptote: x = 1 Horizontal Asymptote/


Www.le.ac.uk Inverse functions Mathematics Department University of Leicester.

to find an Inverse ExamplesIntroduction Question How to find an Inverse ExamplesIntroduction Conclusion Next How to find an Inverse Examples You should now be able to: Calculate inverses. Calculate the range and domain of an inverse. Draw the graph of an inverse function. Introduction


Properties of Functions Section 1.6. Even functions f(-x) = f(x) Graph is symmetric with respect to the y-axis.

function Domain: Range: x-intercept: 0 y-intercept: 0 Odd function Increasing on the interval (-∞, ∞) Cube function x-intercept: 0 y-intercept: 0 The function is neither even nor odd Domain & range nonnegative Increasing on the interval (0, ∞) Minimum value of 0 at x = 0 Square root fx: x-intercept: 0 y-intercept: 0 Domain & range: The function is odd Increasing on the interval (-∞, ∞) No local minimum or maximum Cube root fx: Domain and range/


Unit 3, Lesson 3 Mrs. King  Function rule: an equation that describes a function.  Function notation: when you use f(x) = instead of y =.

. Use 1, 2, 3, and 4 as domain values. t0.5t + 1f(t)f(t) 10.5(1) + 11.5 20.5(2) + 12 30.5(3) + 12.5 40.5(4) + 13 Evaluate the function rule ƒ(g) = –2g + 4 to find the range for the domain {–1, 3, 5}. The range is {–6, –2, 6}./ ƒ(g) = –2g + 4 ƒ(5) = –2(5) + 4 ƒ(5) = –6 ƒ(g) = –2g + 4 ƒ(–1) = –2(–1) + 4 ƒ(–1) = 6 ƒ(g) = –2g + 4 ƒ(3) = –2(3) + 4 ƒ(3) = –2 Find the range of the function ƒ(g) = 3g – 5 for the domain/


Monday, Day 6. Relation – FunctionDomainRange

Monday, Day 6 Relation – FunctionDomainRange – http://youtube.com/watch?v=Rj1sDWjvgjM Your team will create a “Wikipedia Entry” for an assigned astronomer. You will create your “Entry” in Word Be sure to include the following /?v=epJLSCEGjoc Take 10 Minutes 1.The Bus Departs at 8 AM, please plan to be early 2.We are meeting on “The Horseshoe”, in front of the building. 3.We’re going to try and leave GA Tech around noon so we can be back at 1 PM. 4.If you have trouble, Stephen’s Cell is: 770-597-2824 5/


200 300 400 500 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 100 Is it a Function Simplifying Polynomials Adding and.

more points on the graph Give the domain and range of the relation This is what you look for in a mapping diagram to determine whether or not something is a function. *Arrows coming from the DOMAIN: 2 arrows from one number Not a function 1 arrow from each number Function Give the definition of a function. A relation where each element in the domain is paired with exactly one element/


Classifying Relationships.  The definition of a function is:  A function is a relation that maps each element in the domain to one and only one element.

 The definition of a function is:  A function is a relation that maps each element in the domain to one and only one element in the range.  What???  What is domain?  Domain is the “x” values.  What is range?  Range is the “y” values.  So a function in plain English is:  A relation where “x” is not repeated.  There are different ways to determine if a relation is a function depending on/


CSE 2353 – October 1 st 2003 Functions. For Real Numbers F: R->R –f(x) = 7x + 5 –f(x) = sin(x)

f(x) = 7x + 5 –f(x) = sin(x) Function Definition Function from A to B –written f: A  B –a subset of A x B –A is Domain –B is Codomain Mapping or Transformation Equality A = A’ B = B’ F(a) = G(a) for all a  A Image or Range –f: A  B –Image is b  B : f(a/,b)  f for some a  A Composite f(g(x)) f = 2x+3 g = 3x+5 f o g = Terms Injective (No duplicated b’s) Surjective (All b’s used) Graphs Bijective Both Injective and Surjective one to one correspondence /


Hot Seat Game!!! Review of Basic Graphs. Which 3 functions DO NOT have all Real #’s as their domain?

Root Function Which function has its domain as the set of all real numbers except for 0? The Reciprocal Function The Reciprocal Function Which 2 functions have no negative #’s in their domain? The Square Root Function The Square Root Function The Natural Logarithm Function The Natural Logarithm Function Only 3 of the functions are bounded both above and below. Which three? The Sine Function The Sine Function The Cosine Function The Cosine Function The Logistic Function The Logistic Function Three of/


Relations and Functions By: Jeffrey Bivin Lake Zurich High School Last Updated: November 14, 2007.

Relations and Functions By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: November 14, 2007 Definitions Relation  A set of ordered pairs. Domain  The set of all inputs (x-values) of a relation. Range  The set of all outputs (y-values) of a relation. Jeff Bivin -- LZHS Example 1 Relation  { (-4, 3), (-1, 7), (0, 3), (2, 5)} Domain  { -4, -1, 0, 2 } Range  { 3, 7, 5 } Jeff/


2.1 “Relations & Functions” Relation: a set of ordered pairs. Function: a relation where the domain (“x” value) does NOT repeat. Domain: “x” values Range:

2.1 “Relations & Functions” Relation: a set of ordered pairs. Function: a relation where the domain (“x” value) does NOT repeat. Domain: “x” values Range: “y” values Cartesian Coordinate Plane: Definitions Discrete Function: ordered pairs that are NOT connected. Continuous Function: ordered pairs that are connected in a line or a smooth curve. Vertical Line Test: used to determine if the relation is a function. Examples State whether the relation/


Functions Unit Review Take 2 Are you ready?. What is on the Unit Test?

do you tell if a relation is a function? {(6, 3), (7, 4), (9, 3), (6, 3)} What is domain? What is range? Domain and Range? {(6, 3), (7, 4), (9, 3), (6, 3)} How do you evaluate a function? Find f(g(x)), if x = -3 f(x) = -3x + 5g(x) = x 2 – x What are roots/zeros of a function? How do you tell if your approximate/


Acc. Coordinate Algebra / Geometry A Day 28 Agenda: 1. Notes from a PPT: Functions vs. Relations / Domain & Range 2. Homework: Unit 3 Pre-Test Learning.

A Day 28 Agenda: 1. Notes from a PPT: Functions vs. Relations / Domain & Range 2. Homework: Unit 3 Pre-Test Learning Target: Students will know what a function is in comparison to a relation, and how to determine domain & range. Standard: MCC9-12.F.IF.1 and 2 Functions vs Relations Relation Any set of input that has an output Function A relation where EACH input has exactly ONE output Each/


Sequences MM1A1f Recognize sequences as functions with domains that are whole numbers.

. MM1A1f Recognize sequences as functions with domains that are whole numbers. Sequences Sequence-a function whose domain is a set of consecutive whole numbers. Terms-values in the range. Domain-relative position of each term input Range-terms of the sequence output MM1A1f Recognize sequences as functions with domains that are whole numbers. Sequences Write terms of a sequence. a n = 2n-3 Find the first three terms, the domain and range. DomainRange a 1 = 2/


Functions LINEAR AND NON LINEAR. Linear function What is the function that states that the range values are 2 more than the domain values?  f(x) = x.

Functions LINEAR AND NON LINEAR Linear function What is the function that states that the range values are 2 more than the domain values?  f(x) = x + 2 2 xf(x) = x + 2 1 02 13... Linear function – graph is a straight line Linear function What is the function that states that the range values are 1 less than twice the domain/ the coordinates if they are put into a table 4 Non-Linear function What is the function that states that the range values are the squares of the domain values?  f(x) = x 2 5 x f(x)/


PARENT FUNCTIONS Constant Function Linear (Identity) Absolute Value Quadratic Cubic Square Root Greatest Integer Inverse.

Inverse Function or Rational Function (Reciprocal of x) f(x) = 1 x Inverse Function or Rational Function (Reciprocal of x) Parent Equation: Domain: Range: f(x) = 1 x Inverse Function or Rational Function (Reciprocal of x) Parent Equation: x – intercept: y – intercept: Inverse Function or Rational Function (Reciprocal of x) Table: xy -2-0.5 0Error 11 20.5 f(x) = 1 x Parent Equation: Graph Description: Opposite “L” Curves in 1 st and 3/


Www.le.ac.uk Introduction to functions Department of Mathematics University of Leicester.

Functions as graphs Working with functions Intro Question... Functions Mappings Functions as graphs Working with functions Intro What kind of mapping is this? One-to-0neOne-to-manyMany-to-one Conclusion Functions Mappings Functions as graphs Working with functions Next Intro You should now be able to: Find the range and domain of a mapping. Categorise mappings (one-to-one, etc.) Do simple computations with functions. When you feel comfortable move on to composition of functions/


1.4 Functions Function - for every x there is exactly one y. Domain - set of x-values Range - set of y-values Open books to page 40, example 1.

every x there is exactly one y. Domain - set of x-values Range - set of y-values Open books to page 40, example 1. Tell whether the equations represent y as a function of x. a.x 2 + y = 1Solve for y. y = 1 – x 2 For every number we plug in for x, do we get more than / the domain of each function. a.f: {(-3,0), (-1,4), (0,2), (2,2), (4,-1)} Domain = { -3, -1, 0, 2, 4} b. D: c. Set 4 – x 2 greater than or = to 0, then factor, find C.N.’s and test each interval. D:[-2, 2] Ex.g(x) = -x 2 + 4x + 1 Find:a.g(2) /


Section 7.3 Power Functions and Functions Operations.

f (x) = x 2 + 5, find f (3) = f (-8) = f (5280) = f (r) = f (w) = f (BOB) = f (2x – 3) = Composition of functions (a more nerdy version of inception) means we take the equation for g(x) and plug that into f(x). (f composed with g) frac{f(x)}{g(x)} f( Domain: Range: Domain: Range: frac{f(x)}{g(x)} f( Domain: Range: Domain: Range: Try. Quiz next Class


Algebra 4-6 Functions Functions

–4 –2 1 Y X Answer: This relation is not a function because the element 3 in the domain is paired with both 2 and –1 in the range. Answer: This is a function because the mapping shows each element of the domain paired with exactly one member of the range. Example 6-1d Algebra 4-6 Functions Functions You can use the vertical line test to see if a graph/


What are piecewise functions? A __________ function consists of different function rules for different parts of the domain. 1. Graph f(x) = x 2 + 1. 2.

this piecewise function? Explain. Graphing by tables Writing equations of lines xf(x)f(x)xf(x)f(x) Writing an equation of a line If you have slope and y-intercept: If you have slope and a point:/Functions A _________ function is a piecewise function that consists of different constant range values for different intervals of the function’s domain. 1 -2-3 2 -3 -2 3 1 3 2 Rounding-down function Greatest integer function 28.Graph the step function. 29.What is the domain? 30.What is the range? 31.Why is the range/


Derivatives and Inverse Functions 1.Complete the table (without a calculator) for f(x) = sinx 2. Convert both the x- and y- values from your table to the.

the graphs of f(x) = sin x and f -1 (x) = arcsinx 5.Find the values of (f -1 )’(a) if given f(X)=sinx where a=  6.Draw the tangent line to (f -1 )(a) at a=  Domain: Range: Function: y = arcsinxThe derivative formula: Graph: Domain: Range: Function: y = arctanx Graph: Domain: Range: Function: y = arcsecx Graph: The derivative formula: Inverse Trig Functions and Derivatives Domain: Range: Function: y = arccosx Graph: Domain: Range: Function: y = arccotx Graph: Domain: Range: Function: y = arccscx/


In a function every input has exactly one output. Function.

), h(0) h(-3) = h(0) = Give the domain and the range of these functions. Domain: Range: Domain: Range: Give the domain and the range of these functions. Domain: Range: Domain: Range: 12341234 0 InputOutput 10 20 30 40 50 100 200 300 400 500 Input Output What is the domain and range of this table? Domain: Range: Given the function f(x) = 2x + 5 Find the domain and the range of the data InputOutput 3 4 5 6 7 Domain: Range: 12341234 Set ASet B abcdabcd Set ASet B 12341234/


Rational Parent Function Rational Standard Form Example:Example: Transformations: VA: HA: Domain: Range: Y-intercepts: Roots (x-int): VA: HA: Domain: Range:

: Y-intercepts: Roots (x-int): VA: HA: Domain: Range: Example: **Example: Transformations: VA: HA: Domain: Range: What is the domain for f(x) ≥ -7? Y-intercepts: Roots (x-int): VA: HA: Domain: Range: Rational Parent Function Rational Standard Form Example:Example: Transformations: VA: HA: Domain: Range: Y-intercepts: Roots (x-int): VA: HA: Domain: Range: Transformations: VA: HA: Domain: Range: *Go to calculator to graph and get idea of lines between asymptotes – watch out for parenthesis/


2.6 - 1 Continuity (Informal Definition) A function is continuous over an interval of its domain if its hand-drawn graph over that interval can be sketched.

- 2 Example 1 DETERMINING INTERVALS OF CONTINUTIY Describe the intervals of continuity for each function. Solution The function is continuous over its entire domain,(– ,  ). 2.6 - 3 Example 1 DETERMINING INTERVALS OF CONTINUTIY Describe the intervals of continuity for each function. Solution The function has a point of discontinuity at x = 3. Thus, it is continuous over the intervals, (– , 3) and (3,  ). 3 2.6 - 4 Domain: (– ,  ) Range: (– ,  ) IDENTITY FUNCTION  (x) = x xy – 2/


Chapter (3) Transcendental Functions 1- Trigonometric Functions 2- Exponential Functions 3- Hyperbolic Functions.

Functions The exponential function is a function of the form a >0, a ≠ 1 In the definition of an exponential function, a, the base, is required to be positive. Domain: Range: Theorem Example Solution (i) Evaluate the limit (ii) Sketch the graph of the function The Natural Exponential Function The value of e accurate to eight places is 2.71828183. Basic Properties of Natural Exponential Function 3- Hyperbolic Functions The hyperbolic functions are some combinations of and arise/


Functions! By: Haylee Garner. Problem: Kim went to the pumpkin patch. The price for pumpkins were.25 cents per pound. If I picked out a pumpkin that was.

$20. How many pounds was Kim’s pumpkin? Equation Y=.25 or F(x)=.25 Independent and Dependent Variables Independent- number of pounds Dependent- cost Is this a function?? Yes it’s a function because the x values don’t repeat. Domain and Range Domain- The x values (0,1,2,3,4) Range- The y values (0,.25,.50,.75,1) Continuous or Discrete? It’s continuous because/


Relations & Functions An Introduction for Algebra Students.

out this machine! Input Output November 2001 Created by Cathy Stevens7 Think of a relation in terms of input and output Input, x 035035 -2 1 5 6 Output, y November 2001 Created by Cathy Stevens8 The Vocabulary of Relations  X  Input  DomainRange  Output  y November 2001Created by Cathy Stevens 9 A function is a relation in which no two ordered pairs have the same x/


IGCSE FM Domain/Range Dr J Frost Last modified: 14 th October 2015 Objectives: The specification:

, we can’t input negative numbers into our function. The output, again, can only be positive. ? ? Sketch: ? Function Domain Range Function Domain Range Mini-Exercise In pairs, work out the domain and range of each function. A sketch may help with each one. Function Domain Range Function Domain Range Function Domain Range 1 2 3 Function Domain Range 4 Function Domain Range Function Domain Range 8 9 ? ? ? ? ? ? ? 5 Function Domain Range ? 6 7 ? Range of Quadratics A common exam question is to determine the/


CIS 601 – 04 Image ENHANCEMENT in the SPATIAL DOMAIN Longin Jan Latecki Based on Slides by Dr. Rolf Lakaemper.

for general purpose contrast manipulation Transformations Piecewise Linear Transformations Transformations Thresholding Function g(x,y) =L if f(x,y) > t, 0 else t = ‘threshold level’ Piecewise Linear Transformations Input gray level Output gray level Gray Level Slicing Purpose: Highlight a specific range of grayvalues Two approaches: 1. Display high value for range of interest, low value else (‘discard background’) 2. Display high value for/


Which 3 functions DO NOT have all Real #’s as their domain?

Value Function The Reciprocal Function The Reciprocal Function The Squaring Function The Squaring Function Which three functions have NO zeros? The Reciprocal Function The Reciprocal Function The Exponential Function The Exponential Function The Logisitic Function The Logisitic Function Which 3 functions have a range of ? Which 3 functions have a range of ? The Identity Function The Identity Function The Natural Logarithm Function The Natural Logarithm Function The Cubing Function The Cubing Function Which/


Domain x Range y Domain x Range y Domain x Range y 0 1 2 3 n 6 7 8 9 1n + 6 1234n1234n 3 7 11 15 4n - 1 0123n0123n 5 7 9 11 2n + 5 PW page 14 questions.

2 2 2 4 3 6 4 8 Not a function, the red line passes through 2 points during the vertical line test. 2 4 2 4 -2 -4 -2-4 6 8 -6 -8 -6-8 68 Domain Range Input Output Open PW to page 11 and complete problems 3, 4, and 5 One of four parts into which the axes divide a coordinate plane/


EXPONENTIAL FUNCTIONS Section 7.5. 7.5 TOPIC FOCUS I can… Identify exponential growth and decay Graph exponential functions.

AND DECAY? Identify which functions represent GROWTH and which ones represent DECAY RATE YOUR LEVEL OF UNDERSTANDING Got it! Almost There! HELP!!!  MAIN QUESTION DISCOVERY Domain: All Real Numbers HOW DO I GRAPH WITH AN EXPONENTIAL FUNCTION? Example 1 Domain: Range: HOW DO I GRAPH WITH AN EXPONENTIAL FUNCTION? Example 2 Domain: Range: HOW DO I GRAPH WITH AN EXPONENTIAL FUNCTION? Example 3 Domain: Range: HOW DO I GRAPH WITH AN EXPONENTIAL FUNCTION? Example 4 Domain: Range/


8-3 Rational Functions Unit Objectives: Graph a rational function Simplify rational expressions. Solve a rational functions Apply rational functions to.

-world problems Today’s Objective: I can graph a rational function. Rational Function: Hole Asymptote Continuous Graph: No breaks in graph Discontinuous Graph: Breaks in graph V. Asymp: Holes: Domain: Discontinuity: Vertical Asymptotes: Non-removable Discontinuities: Holes: Removable Same factor in numerator and denominator Domain: Where the Denominator = zero Horizontal Asymptotes: No horizontal asymptote Leading term of numerator and denominator (standard form) No horizontal asymptote 1.Find/


Holt McDougal Algebra 2 Rational Functions Holt Algebra 2Holt McDougal Algebra 2 How do we graph rational functions? How do we transform rational functions.

the graph of a rational function can sometimes cross a horizontal asymptote. However, the graph will approach the asymptote when |x| is large. Holt McDougal Algebra 2 Rational Functions Holt McDougal Algebra 2 Rational Functions HA: VA: Graph the function (p < q). State the domain and range. x-intercepts: xy xy Domain: Range: Holt McDougal Algebra 2 Rational Functions HA: VA: Graph the function (p < q). State the domain and range. zeros: xy xy Domain: Range: Holt/


Announcements Starting Unit 7 (Non-Calculator Part) Unit 6 Make-up Test (Pick one of these) –Wednesday: 3:50-5:00pm –Thursday: 7:15-8:30am Unit 6 Reassessment.

. 2 Main Paths –Math/Science/Engineering (Alg3 or Precalculus) –Business (AP Stats or PAP Stats) Square Root Functions Objectives I can graph square root functions using transformations without a calculator I can determine the equation of a square root function from its graph. I can determine domain and range in Interval Notation from a graph Square Root What are critical points Xy 0 1 4 9 16/


7/3/2013  Domain: Set of Inputs ◦ Found on the x-axis  Range: Set of Outputs ◦ Found on the y-axis.

7/3/2013  Domain: Set of Inputs ◦ Found on the x-axis  Range: Set of Outputs ◦ Found on the y-axis Domain  Set-Builder Notation  Interval Notation Range  Set-Builder Notation  Interval Notation  What is domain?  What is range?  What is an intercept (both x and y)?  P. 212 #77-92 The figure below shows the percent distribution of divorces in the U.S. by number of years of marriage. YES! Your temperature/


1.5 Library of Functions Classify functions and their graphs.

1.5 Library of Functions Classify functions and their graphs Linear Functions (a line) Domain: All Realsy = mx + b Range : All RealsAx + By + C = 0 Y intercept (0, b) Writing a linear function Let f(2) = 5; f(4) = 7Points (2, 5);(4, 7) Find the slope y – 5 = 1(x – 2) Writing a linear function Let f(2) = 5; f(4) = 7Points (2, 5);(4, 7) Find the slope/


Unit 10 – Quadratic Functions Topic: Characteristics of Quadratic Functions.

of a quadratic function?  Vertex: (1, 7) o y-value is called maximum o Parabola opens downward (a < 0) Finding domain & rangeDomain: ALWAYS all real #  Range: ALWAYS an inequality –y coordinate of vertex represents minimum or maximum value of rangeRange: y ≥ -6 Finding domain & rangeDomain: all real #  Range: y ≤ 7 What is the axis of/1 and x = 2 What are the zeros of a quadratic function?  one real zero o x = 1 What are the zeros of a quadratic function?  No real zeros Determining a Function /


CHAPTER 2 LESSON 6 Special Functions Vocabulary Step Function- A function whose graph is a series of line segments Greatest Integer Function- A step.

a negative number, the steps go downward instead of upwards, but the rules for step length and distance between steps still apply Examples Constant Function Domain is all real numbers Range is b from f(x)=b Examples Identity Function Identity Function y=x Domain is all real numbers Range is all real numbers Absolute Value Function Domain is all real numbers Range depends on equation Changes that can be made to/


11.2: Graphing Rational Functions Algebra 1: May 1, 2015.

some different features about the graph that are unique from previous functions we have graphed. x-4-20124 y Rational Functions: Graphs Review Domain and Range Domain: x values, input Range: y values, output Vertical asymptote is a value excluded from the domain Horizontal asymptote is a value excluded from the range Find the domain and range of the functions Find the domain and range of the function Graphing: Find all asymptotes – Vertical: Denominator = 0 – Horizontal: If only an/


Trick or Treat Halloween was almost over, and Mr. Green had less than 20 candies left. When the doorbell rang, he thought he would give all the candies.

earnings after 4 weeks. Input (# of weeks) Function rule ( x 5) Output (total earnings) 1 5 x 1 5 2 5 x 2 10 3 5 x 3 15 4 5 x 4 20 Words to Know Domain= the input values Range= the output values In our example the domain would be written as { 1, 2, 3, 4 } and the range would be written as { 5, 10/


~adapted from Walch Education

~adapted from Walch Education Domain and Range ~adapted from Walch Education Concepts: The domain is the set of x-values that are valid for the function. The range is the set of y-values that are valid for the function. A function maps elements from the domain of the function to the range of the function. Each x in the domain of a function can be mapped to one f(x) in the range only. More… To create a mapping, list/


Graphs of Exponential and Logarithmic Functions

Graphs of Exponential and Logarithmic Functions Graph of Exponential Function f ( x ) = bx Domain: ( −∞, ∞ ) or x ∊ ℝ Range: ( 0, ∞ ) or y > 0 Intercept: ( 0, 1 ) y -intercept Asymptote: y = 0 Horizontal Graph of Exponential Function f ( x ) = 2x Domain: ( −∞, ∞ ) or x ∊ ℝ Range: ( 0, ∞ ) or y > 0 Intercept: ( 0, 1 ) Asymptote: y = 0 Now, graph: g ( x ) = 2(x – 3) + 4 Domain: ( −∞, ∞ ) or x ∊ ℝ No change Range: ( 4, ∞ ) or y > 4 Up by 4 Intercept: ( 0/


Week 1 Chapter 1 + Appendix A Signals carry information Signals are represented by functions Systems transform signals Systems are represented by functions,

 1 }. 3 Graph : for each real x, the real sin (x)  [-1,1]. Formally, the graph of a function can be thought of as a set of pairs : { (x,y)  ( Reals  [-1,1] ) | y = sin ( x ) } = { …, (0,0), …, (  /2, 1), …, ( ,0), …, (3  /2,-1), … }. 1 0 If domain and range of a function are finite, then the graph can be given by a table : true true true true false/


Graphs of the Other Trigonometric Functions Section 4.6.

equation. Match a graph to its equation. Determine the domain, range, asymptotes, period and phase shift of the graph of a tangent, cotangent, secant, or cosecant function given a graph. Draw the graphs of a tangent, cotangent, secant, or cosecant function. Determine if the sine, cosine, tangent, cotangent, secant, or cosecant functions are even, odd, or neither. General Function Graph the function tan(x) on the interval [―2π, 2π] What is/


4.7 Inverse Trig Functions

inverse trig functions for specific domains and ranges Evaluated inverse trig functions Evaluated compositions of trig functions 2 Functions that are inverses 2 Functions that are not inverses by evaluating the inner most function first 2 Functions that are not inverses by drawing a triangle Sine Function 1 - 𝜋 2 𝜋 2 -1 Cosine Function 1 π 𝜋 2 -1 Tangent Function - 𝜋 2 𝜋 2 Evaluating Inverse Trig Functions arcTan (- 3 ) Cos −1 (− 3 2 ) arcSin (-1) Composition of Functions When the/


F UNCTIONS AND A BSOLUTE V ALUE Unit One Test Review.

TO … Find the domain and range of a function given its graph Perform operations on functions (add, subtract, multiply, divide) Compose two functions Find and verify inverses Graph absolute value functions Solve absolute value equations and inequalities D OMAIN AND R ANGE F IND THE DOMAIN F IND THE DOMAIN AND RANGE F UNCTION O / X ) = X ²+ 1 G ( X ) = X – 5  Find g(f(x)) I NVERSE F UNCTIONS V ERIFY THAT THE TWO FUNCTIONS ARE INVERSES F(x) = 0.5x + 4 F -1 (x) = 2x – 8 F IND THE INVERSE F UNCTION y = x 2 /


Prerequisite Skills VOCABULARY CHECK 1. The domain of the function is ?. 2. The range of the function is ?. 3. The inverse of the function is ?. ANSWER.

graph at the right. Prerequisite Skills SKILL CHECK Graph the function. State the domain and range. ANSWER 4. y = –2 x – 1 5. y = x + 3 6. y = x – 2 + 5 3 domain: x > 0, range: y < 1 domain: x > 3, range: y > 0 domain: all real numbers, range: all real numbers Prerequisite Skills SKILL CHECK Find the inverse of the function. 7. y = 3x + 5 8. y = –2x 3 + 1 9/


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