Miss DuPree's Math



Unit 2Functions and Their GraphsSection ASetsSet: a collection of unique elementsBy RosterWrite a roster for the set of all integers between 1 and 10- not including 1 or 10.Set Builder NotationReal Numbers:Integers:Natural Numbers:Use set builder notation to write the set of all real numbers, such that, x is less than -4.5Interval NotationNot included/open:Included/closed:Write the following inequality using interval notation: 2≤x<6How do you write a two-part inequality in interval notation?Ex: x<-1 or x>8Write in interval notation:x<0 or 2<x≤10Section 1FunctionsDefinition of a FunctionA function f from a set A to a set B is a rule of correspondence that assigns to each element x in the set A exactly one element y in the set B. The set A is the domain (or set of inputs) of the function f, and the set B contains the range (or set of outputs).Example 1.1Determine if each of the following is a function or not.The input value x is the number of representatives from a state, and the output value y is the number of senators.4572001492250057150010350500y=x2This is the most common way to represent a functionIndependent Variable: Dependent Variable: The domain is the set of all values taken on by _____ and the range is the set of all values taken on by _____.Example 1.2Determine if each of the following is a function of x.a. x2+y=1b. -x+y2=1Try This:Do these equations represent y as a function of x?a. x2+y2=8b. y-4x2=36Function NotationWe know that y=1-x2 gives y as a function of x. We can name this function "f" and then use function notation to describe it:Input: Output:Equation:So y=f(x).Remember f is the name of the function and f(x) is the value of the function at x.To find a function value, substitute the specified input value into the given equation.f(x)=3-2xFind f(-1)F and x are the most commonly used labels for functions, but other letters can be used as well.f(x)=x2-4x+7f(t)=t2-4t+7g(s)=s2-4s+7All these define the same function.Example 1.3Let g(x)=-x2+4x+11. g(2)2. g(t)3. g(x+2)Try This:g(t)=10-3t2a. g(2)b. g(-4)c. g(x-1)Example 1.4Evaluate the function when x=-1, 0, and 1.Example 1.5Find all real values of x such that f(x)=0.a. f(x)=-2x=10b. f(x)=x2-5x+6Example 1.6Find the values of x for which f(x)=g(x).a. f(x)=x2+1 and g(x)=3x-x2b. f(x)=x2-1 and g(x)=-x2+x+2Domain of a FunctionSome domains are described exactly by the function, others are implied. The implied domain is the set of all real numbers for which the expression is defined.f(x)=has an implied domain of all real x other than x=2 and -2.f(x)=√xhas an implied domain of x≥0Example 1.7Find the domain of each function.a. f: {(-3,0), (-1,4), (0,2), (2,2), (4,-1)}b. g(x)=c. Volume of a sphere: V=4/3πr3h(x)=√4-3xExample 1.8You work in the marketing department of a soft-drink company and are experimenting with a new soft-drink can that is slightly narrower and taller than a standard can. For your experimental can, the ratio of the height to the radius is 4.Express the volume of the can as a function of the radius r.Express the volume of the can as a function of the height h.Example 1.9A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and an angle of 45 degrees. The path of the baseball is given by the functiony=-0.0032x2+x+3where y and x are measured in feet. Will the baseball clear a 10-foot fence located 300 feet from home plate?Difference QuotientsExample 1.10Find the difference quotient for f(x)=x2-4x+720574006858000Section 2Analyzing Graphs of FunctionsThe graph of a function f is the collection of ordered pairs (x, f(x)) such that x is in the domain of f. x= the directed distance from the y-axisf(x)= the directed distance from the x-axis331470020955000Example 2.1Use the graph of the function f to find (a) the domain of f, (b) the function values f(-1) and f(2), and (c) the range of f. Vertical Line Test for FunctionsA set of points in a coordinate plane is the graph of y as a function of x if and only if _______________________________________ ______________________________________________________Do these graphs represent y as a function of x?-22860011303000Zeros of a FunctionThe zeros of a function f of x are the x-values for which ________Example 2.2Find the zeros of each function.a. f(x)=3x2+x-10b. g(x)=√10-x2c. h(t)=Increasing, Decreasing, and Constant FunctionsA function f is increasing on an interval if, for any x1 and x2 in the interval, _____________________________________A function f is decreasing on an interval if, for any x1 and x2 in the interval, _____________________________________A function f is constant on an interval if, for any x1 and x2 in the interval, ____________________Example 2.3Use the graphs to describe the increasing, decreasing, or constant behavior of each function.Relative Minimum and Maximum ValuesA function value f(a) is called a relative minimum of f if there exists an interval (x1, x2) that contains a such thatA function value f(a) is called a relative maximum of f if there exists an interval (x1, x2) that contains a such that Example 2.4Use a graphing calculator to find the relative minimum of the function f(x)=3x2-4x-2.Example 2.5Find the relative minimum(s) and relative maximum(s) of the function f(x)=-x3+x.Example 2.6During the 1980s, the average price of a 1-carat polished diamond decreased and then increased according to the modelC=-0.7t3+16.25t2-106t+388, 2≤t≤10where C is the average price in dollars and t represents the calendar year, with t=2 corresponding to January 1, 1982. According to this model, during which years was the price of diamonds decreasing? During which years was the price of diamonds increasing? Approximate the minimum price of a 1-carat diamond between 1982 and 1990.Average Rate of ChangeThe average rate of change between any two points on a graph is the slope of the line through the two points. This line is called the _____________________Example 2.7Find the average rates of change of f(x)=x3-3x froma. x=-2 to x=-1b. x=0 to x=1Even and Odd FunctionsA function is even if, for each x in the domain of f, ____________A function f is odd if, for each x in the domain of f, ____________Example 2.8a. Is the function g(x)=x3-x even or odd?b. Is the function h(x)=x2+1 even or odd?Section 3Parent FunctionsLinear Functionf(x)=mx+bthe domain of the function is the set of ___________________when m≠0, the range of the function is the set of ___________ ______________________the graph has an x-intercept and a y-interceptthe graph is increasing when _________, decreasing when ___________, and continuous when ___________Example 3.1Write the linear function f for which f(1)=3 and f(4)=0.Special Linear FunctionsConstant Function: Identity Function: Squaring Functionf(x)=x2U-shaped curvethe domain is the set of _______________________________the range is the set of ________________________________the function is ______________the graph has an intercept at ___________the graph is decreasing on the interval ______________ and increasing on the interval _______________the graph is symmetric with respect to the ______________the graph has a relative minimum at ______________Cubic Functionf(x)=x3the domain is the set of _______________________________the range is the set of ________________________________the function is ____________the graph has an intercept at _________the graph is increasing on the interval ____________the graph is symmetric with respect to the ____________Square Root Functionf(x)=√xthe domain is the set of _______________________________the range is the set of ________________________________the graph has an intercept at _________the graph is increasing on the interval __________Reciprocal Functionf(x)=1/xthe domain is __________________the range is ___________________the function is __________the graph does not have any interceptsthe graph is decreasing on the intervals __________________the graph is symmetric with respect to the ____________Step functionsFunctions whose graphs represent stairstepsMost famous is the greatest integer function:f(x)=[[x]]=the greatest integer less than or equal to xthe domain is the set of _______________________________the range is the set of ______________________the graph has a y-intercept at ___________ and x-intercepts in the interval ____________the graph is ______________ between each pair of consecutive integer values of xthe graph jumps vertically one unit at each _______________Example 3.2Evaluate the function when x=-1, 2, and 3/2f(x)=[[x]]+1Example 3.3Sketch the graph of:-80010045720000Parent Functions-800100-7620000Section 4Transformations of FunctionsVertical ShiftsExample: f(x)=x2 Shifted up three units.Example: g(x)=√x Shifted down 7 units.Horizontal ShiftsExample: f(x)=x2 shifted right 4 units.Example: g(x)=|x| shifted left 2 units.Vertical and Horizontal ShiftsLet c be a positive real number. Vertical and horizontal shifts in the graph of y=f(x) are represented as follows:1. Vertical shift c units upward: 2. Vertical shift c units downward: 3. Horizontal shift c units to the right: 4. Horizontal shift c units to the left: Example 4.1Describe the graph of the following functions in relationship to the graph of f(x)=x3a. g(x)=x3+1b. h(x)=(x-1)3c. k(x)=(x+2)3+1274320022161500Example 4.2Each of the graphs shown is a vertical or horizontal shift of the graph of f(x)=x2. Find an equation for each function.ReflectionsReflections in the coordinate axes of the graph of y=f(x) are represented as follows:1. Reflection in the x-axis: 2. Reflection in the y-axis: 2628900-57150000Example 4.3This is the graph of f(x)=x4Use this to write the equation of each of these functions:1143008382000Example 4.4Compare the graph of f(x)=√x with each of the following:a. g(x)=-√xb. h(x)=√-xc. k(x)=-√x+2Nonrigid TransformationsWhat is a nonrigid transformation?If a nonrigid transformation on the graph of y=f(x) is represented by y= cf(x)If c>1 it is a If 0<c<1 it is a If a nonrigid transformation on the graph of y=f(x) is represented by y=f(cx)If c>1 it is a If 0<c<1 it is a Example 4.5Compare the graph of each function with the graph of f(x)=|x|a. h(x)=3|x|b. g(x)=(1/3)|x|Example 4.6Compare the graph of each function with the graph of f(x)=2-x2a. g(x)=2-8x3b. h(x)=2-(1/8)x3Section 5Combinations of Functions and Composite FunctionsArithmetic Combinations of FunctionsJust like with real numbers, two functions can be combined by addition, subtraction, multiplication, and division to form new functions.f(x)=2x-3g(x)=x2-1Find the sum, difference, product, and quotient of f and g.Example 5.1Find (f+g)(x) for the functionsf(x)=2x+1 and g(x)=x2+2x-1Then evaluate the sum when x=3 -- is this value the same as evaluating f(3)+g(3)?Example 5.2Find (f-g)(x) for the functionsf(x)=2x+1 and g(x)=x2+2x-1Then evaluate the difference when x=2.Is this value the same as f(2)-g(2)?Example 5.3Given f(x)=x2 and g(x)=x-3 find (fg)(x). Then evaluate the product when x=4.Example 5.4Find (f/g)(x) and (g/f)(x) for the functionsf(x)=√x and g(x)=√4-x2Then find the domains of f/g and g/fCompositions of Functionsf(x)=x2 and g(x)=x+1The composition of f with g is:f(g(x))=f(x+1)=(x+1)2Also denoted f°gExample 5.5Given f(x)=x+2 and g(x)=4-x2, find:a. (f°g)(x)b. (g°f)(x)c. (g°f)(-2)Example 5.6Find the composition (f°g)(x) for:f(x)=x2-9 and g(x)=√9-x2Then find the domain of (f°g)***Note the domain of f and g individually!Function “Decomposition”We need to be able to identify two functions that make up a given composite function.h(x)=(3x-5)3What two functions were combined to create h(x)?Example 5.7Express the functionh(x)=as a composition of two functions.Example 5.8The number of bacteria in a refrigerated food is given byN(T)=20T2-80T+500, 2≤T≤14where T is the Celsius temperature of the food. When the food is removed from refrigeration, the temperature is given byT(t)=4t+2, 0≤t≤3where t is the time in hours. Find:The composite N(T(t)). What does this function represent?The number of bacteria in the food when t=2 hours.The time when the bacteria count reaches 2000.Section 6Inverse FunctionsInversesInverse functions have the effect of "undoing" each otherInverse of f(x) is denoted by ___________The domain of f is the _____________The range of f is the _____________The composition of f with f-1 (and that of f-1 with f) is the ________________________For Example:If f(x)=x+4, then f-1(x)=f(f-1(x))=f-1(f(x))=f(x)=x+4={(1,5), (2,6), (3,7), (4,8)}f-1(x)=x-4={(5,1), (6,2), (7,3), (8,4)}Domain of f(x):Range of f(x):Domain of inverse:Range of inverse:Example 6.1Find the inverse of f(x)=4x. Then verify that both f(f-1(x)) and f-1(f(x)) are equal to the identity function (both equal x when simplified).Definition of InverseLet f and g be two functions such thatf(g(x))=x for every x in the domain of gand g(f(x))=x for every x in the domain of f.Under these conditions, the function g is the inverse of the function f. The function g is denoted by f-1 (read "f-inverse"). Thus,f(f-1(x))=xandf-1(f(x))=x.The domain of f must be equal to the range of f-1, and the range of f must be equal to the domain of f-1.Example 6.2Show that the functions are inverses of each other:Example 6.3Which of the functions is the inverse of f(x)=5/(x-2)?The Graph of an Inverse FunctionThe graphs of a function f and its inverse function f-1 are related to each other in the following way.If the point (a, b) lies on the graph of f, then the point (b, a) must lie on the graph of f-1, and vice versa.This means that the graph of f-1 is a reflection of the graph of f in the line y = x.The Existence of an InverseConsider f(x)=x2We would assume the inverse to be f-1(x)=_____________But, according to the rules of inverses, the domain of f must be the range of f-1. In this case, the domain of f is ___________________, and the range of f-1 is ____________________.To have an inverse, a function must be one-to-one.One-to-one: For every a and b in the domain of f, f(a)=f(b) implies that ______________ This means that each x is mapped to exactly one y and each y is mapped to exactly one x.Example 6.4Look at the graph of each function to determine whether or not the function has an inverse function.a. f(x)=x3-1b. g(x)=x2-125146009398000-1143009398000Example 6.5Find the inverse (if it exists) off(x)=Example 6.6Find the inverse of f(x)=√2x-3Example 6.7Find the inverse of:a. f(x)=x+4b. g(x)=2x-1c. h(x)=-3x+1Example 6.8Find the inverse of each:a. f(x)=x6, x≥0b. g(x)=-x3+4c. h(x)=1/27 x3d. k(x)=2x5+3Example 6.9Your hourly wage is $7.50 plus $0.90 for each unit x produced per hour. Let f(x) represent your weekly wage for 40 hours of work. Does this function have an inverse? If so, what does the inverse represent?Example 6.10Let x represent the retail price of an item (in dollars), and let f(x) represent the sales tax on the item. Assume that the sales tax is 7% of the retail price AND that the sales tax is rounded to the nearest cent. Does this function have an inverse? (Hint: Can you undo this function? For instance, if you know that the sales tax is $0.14, can you determine EXACTLY what the retail price is?) ................
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