INTRODUCTION TO GRAPHS AND FUNCTIONS Introductory Topics in Graphing Distance Between Two Points On The Coordinate Plane.

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1 P ( 1, 1 ) INTRODUCTION TO GRAPHS AND FUNCTIONS Introductor Topics in Graphing Distance Between Two Points On The Coordinate Plane Q (, ) R (, 1 ) The straight line distance between two points P ( 1, 1 ) and Q (, ) is the hpotenuse of a right triangle whose sides are the horizontal and vertical distances. (Distance between P and Q) = hpotenuse = (horizontal side) + (vertical side) = ( 1 ) + ( 1 ) Take the square root of both sides to find the distance d: Distance d = ( 1) + ( 1) The midpoint of the line segment between two points P ( 1, 1 ) and Q (, ) Midpoint =, 1. For the points P and Q in the graph above, find the distance between the points and find the midpoint of the line segment connecting points P and Q.. For points A (5, 3) and B ( 1, ) find the distance between the points and the midpoint of the line segment connecting the points. Circles On The Coordinate Plane A circle is the set of points that are equidistant from a point (h,k) which is the center of the circle. The radius is the distance from the center to an point on the circle. We can use the distance formula to find the equation of a circle. (P. 0 in tet; shown in class) The equation of a circle with radius r and center (h,k) is ( h) + ( k) = r 5. Find the equation of the circle a. with radius 5 and center (1,) b. with radius 8 and center (0, 3) 6. Find the center and radius of the circle whose equation is ( 3) + = Find the center and radius of the circle whose equation is ( + 4) + ( 4 7 ) = 1 8. For the circle ( 3) + = 81, solve for and write the equation in " = " form. Rewrite it as two separate equations and eplain what each part of the equation represents. Page 1

2 INTRODUCTION TO GRAPHS AND FUNCTIONS Introductor Topics in Graphing: Smmetr We can visualize the graphs of equations b plotting points on the plane. In a few das we will use our graphing calculator to graph equations. The graphing calculator plots points and then "connects the dots". Usuall this gives reasonabl correct graph of the equation. Sometimes the graphing calculator does not connect dots in an appropriate manner and the graph is not correct. We need to be able to visualize what the graph of an equation looks like. We will use the graphing calculator to help us, but we must also use our critical thinking skills to decide whether the graphing calculator is showing the correct graph. For each of the equations below, plot the points and then connect the points. For the top two graphs, use straight lines to connect the points. For the bottom two graphs, use smooth curves to connect the points. = + = = = * turning point relative maimum * turning point relative minimum Page

3 The graphs of = + and = 4 ehibit smmetr about the -ais INTRODUCTION TO GRAPHS AND FUNCTIONS Introductor Topics in Graphing: Smmetr The graph of = 3 4 ehibits smmetr about the origin The graph of = 3 ehibits smmetr about the -ais See page 18 in tetbook for more eamples of these tpes of smmetr. Smmetr with respect to -ais Smmetr with respect to -ais Smmetr with respect to origin Graphical Tests for Smmetr (page 18) whenever the point (,) is on the graph, the point (, ) is also on the graph whenever the point (,) is on the graph, the point (, ) is also on the graph whenever the point (,) is on the graph, the point (, ) is also on the graph Algebraic Tests for Smmetr (page 19) replacing with ields an equivalent equation replacing with ields an equivalent equation replacing both with and with ields an equivalent equation Use the algebraic test to determine the tpe of smmetr, if an, that applies to each of the following: (without looking at the graph) Test for smmetr about ais: replace b = 4 3 = 5 + = = about ais : replace b about origin replace b and replace b Page 3

4 INTRODUCTION TO FUNCTIONS Each of these eamples has an input set and an output set. The arrows show which numbers in the input set are related to which numbers in the output set. For each eample, describe how the numbers in the input set are related to the numbers in the output set. Eample 1. Eample. Input Output Input Output Describe how the output numbers are related to the input numbers b the arrows Describe how the output numbers are related to the input numbers b the arrows Eample 3. Eample 4. Input Output Input Output Describe how the output numbers are related to the input numbers b the arrows Describe how the output numbers are related to the input numbers b the arrows Suppose ou had a TRANSFORMATION MACHINE. When ou put an input number in, it is transformed into an output number. In each eample above, tell whether ou can reliabl predict what number will come out when ou put an input number into the transformation machine. IN OUT The eamples where each input can have eactl one output are called functions. Eamples 3 and 4 are functions. When ou input a number, ou can reliabl predict which number will be the output. Page 4

5 Eample 4: Absolute Value Function OUT When is input, we know that will be its output. In eamples 1 &, each input has more than one possibilit for its output. When ou input a number, ou can not predict with certaint which number be the output. Eamples 1 & are relations, but the are not functions. Eample : Less Than Relation (NOT a function) 4 OUT??? When we input "4", we don't know whether the output will be 1,, or 3. The definition of a function in our tetbook page 40: A function f from a set A to a set B is a relation that assigns to each element in the set A eactl one element in the set B. Definition of function in everda language: Ever input has eactl one output If an input has more than one possible output, the relation is NOT a function We can write the relations and functions as sets of ordered pairs instead of using diagrams with arrows. Ordered Pair = (input, output) = (,) Look back at the tables for each eample and write the ordered pairs below. Eplain how ou can see from the ordered pairs which is a function and which is not. Eample 1: Eample : Eample 3: Eample 4: Was in which functions can be represented: Some functions can be represented in all these was. Some functions can onl be represented in some of these was. As sets of ordered pairs As a table As a formula As a graph As a description (sentence form) Mathematical description: Multipl the input b to obtain Table : input : output the output value 1 Formula = 0 0 Ordered pairs 1 ( 1, ), (0,0), (1, ), (,4) 4 Verbal description, modeling a real life situation: De Anza College recommends that students stud hours outside the classroom for ever hour spent in class. = input = number of hours spent in class = output = number of hours of stud needed outside of class. (Note that = 1 would not be valid input for this situation) Graph for the given values of from the table Graph, if the domain includes all real numbers Page 5

6 Domain : Set of INPUT values Range: Set of OUTPUT values Each number that is legal input (in the domain) to the function will be associated with eactl one number that is its reliable, consistent, predictable output. Identifing a Function from a Formula: Step 1: Solve the equation to put it in = form: Step : If each value will give onl one value, then it is a function If some value(s) will give more than one value, then it is NOT a function + = 1 = 1 + = 1 = 1 Identifing Functions from Graphs Vertical Line Test Page 55 A set of points in a coordinate plane is the graph of as a function of if and onl if no vertical line intersencts the graph at more than one point. A graph represents a function if whenever ou draw a vertical line through an value in the domain, it intersects the graph onl once. If an vertical line crosses the graph more than once, it is not a function A. B. C. D. E. F. G. H. Which of the graphs represent functions? PRACTICE: For each graph that represents a function, identif the domain and the range from the graph. Assume that the graph continues off in the same direction at the edges of each graph Page 6

7 Working with Function Notation Function notation: shows the input and the output: f(s) = s Freida s Fudge Factor produces chocolate fudge that is 1 inch thick (height) and is cut into square pieces. This function describes the volume of a square piece of 1 inch thick fudge if the side of the square is inches. Write the function notation that describes the situation. Substitute into the formula and simplif or evaluate. a. The volume of a square piece of fudge with side of length inches b. The volume of a square piece of fudge with side of length 3 inches c. The volume of a square piece of fudge with side of length s inches d. The volume of 3 square pieces of fudge with side of length s inches e. The volume of a square piece of fudge with side of length s inches f. The volume of a square piece of fudge with side of length s + inches g. The volume of two square pieces of fudge, one piece with side length s and the piece other with side length t h. 100 cubic inches of fudge plus 1 more pieces of fudge each with side of length inches. Practice Problem: Do this problem for homework Suppose the hemispherical dome of the planetarium needs to be painted. The surface area of a hemisphere with radius r is (1/)(4πr ) = πr A paint manufacturer estimates that 1 gallon of paint covers 400 square feet of area. The number of gallons of paint needed to paint a hemispherical dome of radius r feet is f(r) =πr /00 Write the function notation that describes the situation. Substitute into the formula and simplif or evaluate. a. The amount of paint needed to paint De Anza s planetarium dome (outdoor radius is appro. 30 ft.) b. The amount of paint needed to paint coats of paint on De Anza s planetarium dome. c. The amount of paint needed to paint a dome whose radius is twice as large as De Anza s dome. d. The amount of paint needed to paint a dome with radius r + 10 ft. e. The amount of paint needed to paint a dome with radius r f. The amount of paint needed to paint 3 coats of paint on a dome with radius r g. The amount of paint needed to paint domes, one with radius 30 feet and another with radius 40 feet. Page 7

8 INTRODUCTION TO GRAPHS AND FUNCTIONS: Finding the Domain of a Function The domain is the set of input values for the function. When we are asked to find the domain of a function, we want to find the values of the independent (input) variable that are "legal" to input into the function. Eample 0: f() = + 3 Domain : Eample 1: f() = 1 Domain : Eample : f() = Domain : Eample 3: f() = Domain : Eample 4: f() = + 3 Domain : Eample 5: f() = 1 Domain : Eample 6: f() = 0 + Domain : Sometimes a function has more than one restriction that affects its domain Eample 7: f() = Eample 8: f() = Eample 9: f() = Domain : Domain : Domain : Eample 10: f() = 4 4 = 3 4 ( + 1)( 4) Domain : Eample 11: f() = 3 3 = 3 4 ( + 1)( 4) Domain : Eample 1:{ f() = log Domain : Check for even roots: the radicand (or base if in eponential form) can not be negative for even roots, but can be negative for odd roots. Check for division b 0: Set the denominator = 0 and solve for. EXCLUDE those values from the domain. Check for log functions: The input into a log function must be positive Page 8

9 INTRODUCTION TO GRAPHS AND FUNCTIONS: Finding the Range of a Function The range is the set of output values for the function. When we are asked to find the range of a function, we want to find all the values of the dependent (output) variable of that can be obtained as output from the function. Eample 1: = f() = Eample : = f() = Eample 3: = f() = + Range : Range : Range : Eample 4: = f() = 3 Range : Eample 5: = f() = Range : Eample 6: = f() = sin Range : Smmetr of Functions: Even and Odd Functions Graphical Algebraic A function is an even function if: its graph is smmetric about the ais f( ) = f() whenever is replaced b A function is an odd function if: its graph is smmetric about the origin f( ) = f() whenever is replaced b Note that if a graph is smmetric about the ais, it is not the graph of a function. For each graph, is it an even function, odd function, a function but not odd or even, or not a function? = + = 3 4 = 3 = 4 = (+) You should be able to determine smmetr b looking at the graph AND also be able to show it algebraicall without looking at the graph. 4. Algebraicall, determine whether each of the following functions is even, odd,or neither. = f() = 4 3 = g() = 5 + = h() = Page 9

10 INTRODUCTION TO GRAPHS AND FUNCTIONS: Describing the Behavior of a Function f is increasing if f(a) < f(b) whenver a < b f is decreasing if f(a) > f(b) whenver a < b f is constant if f(a) = f(b) for all a and b Where a function is shaped like or part of a, (or a smile) it is concave up. Where a function is shaped like, or part of a (or a frown) it is concave down. A function has a relative maimum = f(a) at the value = a when there is no point nearb = a that has a larger value. On a graph, this generall is the top of a hill. A function has a relative minimum = f(a) at the value = a when there is no point nearb = a that has a smaller value. On a graph, this generall is the bottom of a valle. (RULE: an endpoint can't be a relative minimum or a maimum) When a question asks "on what intervals" ou need to answer with the appropriate values, taking into account the fact that is continuous and is not restricted to integer values. For the function in the graph at the right, on what intervals for is the function: Increasing? Decreasing? Constant? Concave up? Concave down? Page 10

11 INTRODUCTION TO GRAPHS AND FUNCTIONS Piecewise Functions Different equations (formulas) are needed to represent different parts of the function Absolute Value Function = f() = f() = < < 1 g() = 1 < (1/) Step Functions It costs $.5 (5 cents) a minute to talk on a prepaid cell phone plan Draw the graph: The cost of parking at an airport is $1 for the first hour or an part of the first hour, and $ for each additional hour or an part of the additional hour. Draw the graph: The "greatest integer" function: the largest integer that is less than or equal to = f() = int() = int() = int(.6) = int(π) = int(0) = int( 3) = int( 5.4) = Page 11

12 INTRODUCTION TO GRAPHS AND FUNCTIONS: Average Rate of Change The average rate of change is the SLOPE of the LINE segment that connects two points that lie on the graph of a continuous function. Continuous means ou can trace the graph of the function without picking up our pencil from the paper; the graph has no gaps or jumps. The graph itself ma be a line or a curve. Eample 1: The forest service introduced 00 fish into a lake. At the end of 6 ears, there are 650 fish. Below are graphs of 3 possible models for how the fish population grew from 00 to 650 fish. a. For the graphs of f and g draw a smooth curve through the points without lifting our pencil. Compare the three graphs. Describe in words how the fish population grows over time in each graph. F I S H F I S H F I S H ears ears ears L() = f() = g() = b. On each graph, draw a line that connects the point where = 0 to the point where =. Because L() is a line, ou have traced part of the line, between = 0 and =. Because f() and g() are curves, ou have drawn a line that connects two points on the curve, but does not follow the curve. The line segment is called a secant line. c. For each graph above, using the (,) coordinates for = 0 and =, find the slope of the line segment. L: f: g: The slope of a secant line connecting two points on the graph of a continuous function is called the average rate of change. 1 The slope of the secant line is, where ( 1, 1 ) and (, ) are 1 the endpoints of the secant line on the graph of the function = f(). We usuall use function notation f() instead of : average rate of change = f() f(1) If the function is represented b a different letter, such as g, use that letter instead of f. Page 1 1

13 f() f(1) 1 = Average Rate of Change of a function f() from = 1 to = Eample 1: (continued) Draw the curves for functions f and g. d For the time period from = to = 6 ears, draw the secant lines and calculate the average rate of change. F I S H F I S H ears ears ears L() = f() = g() = L: f: g: F I S H Eample : Evaluate =f() = f() = using a table on our calculator 30 and fill in below Draw the secant lines and calculate the average rate of change (a) from = 0 to = 1 (b) from = to = 4 (c) from = 5 to = 7 (d) from = 7 to =8 When the function is increasing, the sign of the average rate of change is When the function is decreasing, the sign of the average rate of change is When the function is concave up, the secant lines lie the graph of the function. When the function is concave down, the secant lines lie the graph of the function. Page 13

14 INTRODUCTION TO GRAPHS AND FUNCTIONS: Difference Quotient and Average Rate of Change Suppose we are at the point (, f()) on a graph and want to move to a nearb point. This nearb point is h units awa from along the ais. Its -coordinate will be + h, so its -coordinate will be = f(+h)). If we consider our original point (,f()) as ( 1, 1 ) and our new point (+h, f(+h)) as (, ), then the average rate of change formula becomes average rate of change = f( ) f( 1 ) f( + h) f() f( + h) = = ( + h) h 1 f() This formula is also called a difference quotient. difference quotient = average rate of change = f( + h) f() h Eample: For the function = f() = 7, find and simplif the difference quotient formula. Then evaluate the formula for = 6 and h = 0.5 Page 14

15 Librar of Functions Graph the following equations b hand or using our graphing calculator. If using the calculator, use the standard graphing window: 10 10, ; Scl and Scl = 1 TI 8, 83, 84 ZOOM 6:Standard TI-86 GRAPH F3:ZOOM F4:ZSTD TI:83&84 Find abs and int on the MATH NUM menu TI:86 Find abs and int on the nd Math F1:NUM menu Sketch the graph of each functions on the worksheet below. Save this for reference. Stud, learn & remember the shapes of these different basic tpes of functions. Linear Functions & Equations Absolute Value Function = c = = = c = = abs() Constant Function Identit Function (not a function) Even Integer Powers of = = 6 Quadratic Function Odd integer powers of = 3 = 5 Cubic Function Even Roots of = 1/ = = 1/4 = 4 Square Root Function Odd Roots of = 1/3 = 3 = 1/5 = 5 Cube Root Function Reciprocal Function = 1 1 = More Negative Powers of = 1 = = 3 1 = 3 Step Function = = int() Page 15

16 Transformation of Functions (Refer also to Transformations Homework Worksheet) To find the graph of f() c, c > 0, f() +c, c > 0 f( c), c > 0, f(+c), c > 0 cf(), 0 < c < 1 cf(), c > 1 f(c), 0 < c < 1 f(c), c > 1 f() f( ) Describe the transformation of the graph of = f() When doing more than one transformation, order matters. Not following the correct order ma give incorrect results Vertical Transformations: Reflections and stretches/compressions have first priorit and can be done in an order. Do vertical shifts up or down after vertical reflections, stretches or compressions. Horizontal Transformations: Factor the epression inside the function. Eamples: (c d) = d c ( 6) = (( 3)) c First shift function horizontall b d/c units (right if d/c > 0, left if d/c < 0) Net do horizontal stretches/compressions using the line = d/c as the ais of reference Then do horizontal reflections using the line = d/c as the ais of reference For each problem the graph of = f() is shown. Describe the transformations of the graph of = f() to obtain the given function. Then use the transformations to draw the graph of the given function. g() = 3f(+) h() = 3f( 4) s()=.5f(.5) 3 Describe the transformations of the graph of = to obtain the given function. Then use the transformations to draw the graph of the given function. = u() = +3 = v() = +3 = w() = 6 Page 16

17 Transformations of Functions: Finding Equations From Graphs A B (-,-7) (,-7) (-1,-4) Equation (1,-4) Equation C D Equation Equation E F Equation Equation Page 17

18 INTRODUCTION TO GRAPHS AND FUNCTIONS: Algebraic Combination of Functions Functions can be added, subtracted or multiplied A function can be divided b another function, when the value of the function in the denominator is not 0. The domain of the arithmetic combination of functions f and g is the set of all numbers that are common to BOTH domains EXCEPT if the arithmetic combination is a quotient (division). For a quotient of functions, values of that make the denominator equal to 0 must be removed from the domain. For the functions f() = and g() =, find the sums, differences and products, and quotients: Arithmetic Combination of Functions ( f+g ) () = f() + g() = ( f g ) () = f() g() = ( fg ) () = f()g() = Domain f () g g () f For the functions f() = 3+ and Arithmetic Combination of Functions ( g f ) () = g() f() = g() =, find the combinations requested: Domain ( fg ) () = f()g() = f () g g () f For the functions f and g shown in the graph, find (f+g)() (f g) () (fg) ( 3) f (3) g f g (1) (1) g f f g Page 18

19 INTRODUCTION TO GRAPHS AND FUNCTIONS: Composition of Functions A composition of functions is when we use the output of one function as the input to another function Order is Important!: Alwas work "from the inside to the outside" or "from right to left" f() = g() = + 1 Find f g () = f(g()) Find g f () = g(f()) Evaluate f g ( ) Evaluate g f ( ) Composition is NOT the same as multiplication; the usuall do not give the same results Find the product (fg) () = f()g() The product obtained b multipling the functions is not the same as the compositions above. f() = + 1 g() = 9 f g () = f(g()) g f () = g(f()) Application 1 You are going to bu a desk at an office suppl store. You have coupons, one for $5 rebate for a purchase of an desk; the other for 0% of our entire purchase. a. If onl the $5 rebate is given, epress the sale price as a function of : = R() = If onl the 0% discount is given, epress the sale price as a function of : = D() = b. Suppose the store is willing to give both the rebate and the discount. You want to know if it is better for ou, the consumer, if the store calculates the 0% discount first or the $5 rebate first. Assume the desk costs $00. Find cost for each situation: $5 Rebate Onl: 0% Discount Onl $5 Rebate first, then 0% Discount 0% Discount first, then $5 Rebate c. For each situation below, epress the sale price using a composition of functions R() and D() for a desk with a list price of $ $5 Rebate first, then 0%Discount = 0%Discount first, then $5 Rebate: = Page 19

20 COMPOSITION OF FUNCTIONS Application You are on a boat in Lake Tahoe and accidentall drop our sunglasses into the lake. At the spot where our glasses hit the water, a small wave forms, and concentric circular waves begin to propagate out from the spot where our glasses fell in. Suppose that the radius of the circle increases at the rate of 5 inches per second. a. r(t) represents the radius as a function of time t (assume r(0) = 0) : r(t) = b. A(r) represents the area of the circle as a function of the radius r : A(r) = c. Find the composition of functions that represents the area as a function of time. d. Find the area at t = 3 seconds. Decomposing functions: Each function below can be written as a composition of two functions f g(). Specif f and g. BE CAREFUL ABOUT THE ORDER! You must choose functions f, g, so that f(), g() U() = + 8 T() = cos 3 W() = ( + 3 +) 6 Z() = +3 Write each of the following functions as a composition of three functions f g h BE CAREFUL ABOUT THE ORDER! You must choose functions f, g, h so that f(), g(), h() M() = tan( 4) 3 + P() = ( ) C() = cos 3 (+1) D() = 1 sin Distinguishing Products and Compositions: A product of functions is obtained when we multipl two functions b each other. A composition of functions is when we use the output of one function as the input to another function. Which of the following is a product of functions? Which of the following is a composition of functions? a. H() = sin J() = sin ( ) b. Q() = R() = (4 1)10 c. S () = cos (sin ) T () = cos sin d. U() =3e V() = e 3 Page 0

21 COMPOSITION OF FUNCTIONS: Graphical Interpretation Using the functions given in the graph, find (f g) ( 1) f g (g f) () (f g) (3) (es it s a trick question ) COMPOSITION OF FUNCTIONS: Domain To find the domain of a composition of functions (f g): (f g)() = f(g()) use this 3 step process: Step 1: Find the domain (set of legal input) for g Step : Then find the outputs from g that are not legal inputs to f Find the value(s) of that is/are input to g to create that output Step 3: Eliminate those values of from the domain of g to get the domain of (f g) When ou compose two functions, it is important to specif the domain of the composition, using this method. Warning: If ou just find the formula for f(g()) and do not use this process to find the domain, ou ma get in incorrect answer for the domain of f(g()) Eample A: Eample 5 in tet : f() 9 g() = 9 : Find the domain of f(g()) Step 1: Domain of g is 3 3 Step : All output values from g are legal input to f Step 3: The domain of f(g()) is 3 3 [ 3, 3 ] If ou just found the formula for f(g()) = ou would incorrectl think the domain is all real numbers 1 Eample B: f() = Step 1: Domain of g is 100 and g() = 100 : Find the domain of f(g()) Step : BUT when g has output g() = 0, then ou can't find f() because of division b 0 The output g() = 0 occurs when = 100 Step 3: Therefore = 100 must be eliminated from the domain The domain of f(g()) is > 100 (100, ) 1 If ou just found the formula for f(g()) = 100 ou would incorrectl think the domain is all real numbers ecept 100 Page 1

22 INTRODUCTION TO GRAPHS AND FUNCTIONS: One to One Functions A function is ONE TO ONE if there is one value for ever value. You can trace an value back to its corresponding value, and no other value has the same. This corresponds to a HORIZONTAL LINE TEST A graph represents a ONE TO ONE FUNCTION if whenever ou draw a HORIZONTAL line through an value in the domain, it intersects the graph onl once. If an horizontal line crosses the graph more than once, it is NOT a ONE TO ONE function Note that it must also satisf the vertical line test to be a function: A graph represents a function if whenever ou draw a vertical line through an value in the domain, it intersects the graph onl once. If an vertical line crosses the graph more than once, it is not a function = f() = g() = s() = u() = w() INTRODUCTION TO GRAPHS AND FUNCTIONS: Inverse Functions Intuitive Definition of Inverse Function An inverse f 1 of a function f "undoes" f, bringing ou back to the original input Condition For Eistence Of An Inverse Function A function must be a one to one function in order to have an inverse. (If a function is not one-to-one, then there are some values that are output from more than one -value as input.) Which functions shown above have inverse functions? Which functions shown above do NOT have inverse functions? Give a numerical eample that eplains wh = f() = does not have an inverse function: Mathematical Definition of Inverse Function If a function f has an inverse f 1, then f 1 is the function for which f 1 f () = AND f f 1 () = Eample: f() = f 1 7 () = 1 Show that these functions are inverses of each other using composition: (SHOW YOUR WORK!) f 1 f () = f 1 (f ()) = f f 1 () = f (f 1 ()) = Page

23 INTRODUCTION TO GRAPHS AND FUNCTIONS: Inverse Functions Graphical Eample: Find g 1 (3) Find g 1 () g Find (g 1 )(-1) Find g 1 (0) Ordered Pairs: If ou represent a function as ordered pairs, the roles of and are switched when ou write the inverse function as ordered pairs. f() = 3 {( 3, 7), (, 8), ( 1, 1),(0,0), (1,1), (,8), (3,7)} f 1 () = 1/3 = 3 { } Domain and Range: The domain of the function = f() becomes the range of the inverse = f 1 () The range of the function = f() becomes the domain of its inverse = f 1 () Graphical Smmetr Graph f() = 3 and f 1 () = 1/3 = 3 Accuratel plot the points for =, 1, 0, 1, and connect them with a smooth curve to represent the functions. Use different colors. Draw the line =. The graph of f 1 is the reflection of the graph of f across the line =. Draw the line = and use it as a guide to help draw the graph of g 1 g Domain of g: Range of g: Domain of g 1 : Range of g 1 : Page 3

24 INTRODUCTION TO GRAPHS AND FUNCTIONS: Inverse Functions Finding the inverse function algebraicall: 1. First determine whether f has an inverse (Is f one-to-one? Horizontal Line Test). In the equation for f(), replace f() b 3. Solve for = 4. Interchange the letters and 5. Replace b f 1 () in the new equation 6. Verif that f and f 1 are inverses using composition This is NOT the same order as the instructions in the tetbook, but will lead to the same result. Eamples: Each of the following functions is one-to-one and has an inverse. Find the inverse function for each A. f() = (p. 97) D. g() = (p. 99, #4) B. f() = E. P() = 75, 100 C. f() = (p. 98) Page 4