INTRODUCTION TO GRAPHS AND FUNCTIONS Introductory Topics in Graphing Distance Between Two Points On The Coordinate Plane y Q (x2, y2) P (x1, y1) R (x2, y1) x The straight line distance between two points P (x1, y1) and Q (x2, y2) is the hypotenuse of a right triangle whose sides are the horizontal and vertical distances. (Distance between P and Q)2 = hypotenuse 2 = (horizontal side) 2 + (vertical side) 2 = (x2 − x1) 2 + (y2 − y1) 2 Take the square root of both sides to find the distance d: Distance d = The midpoint of the line segment between two points P (x1, y1) and Q (x2, y2) (x2 −x1)2 +(y2 − y1)2 x1 + x 2 y1 + y 2 , 2 2 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. 2. For points A (5, −3) and B ( −1, 2) 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 any point on the circle. We can use the distance formula to find the equation of a circle. (P. 20 in text; shown in class) The equation of a circle with radius r and center (h,k) is 5. Find the equation of the circle a. with radius 5 and center (1,2) (x − h) 2 + (y − k) 2 = r2 b. with radius 8 and center (0, −3) 6. Find the center and radius of the circle whose equation is (x − 3) 2 + y 2 = 81 7. Find the center and radius of the circle whose equation is (x + 4) 2 + ( y – 7 2 ) = 12 4 8. For the circle (x − 3) 2 + y 2 = 81, solve for y and write the equation in " y = " form. Rewrite it as two separate equations and explain what each part of the equation represents. Page 1 INTRODUCTION TO GRAPHS AND FUNCTIONS Introductory Topics in Graphing: Symmetry We can visualize the graphs of equations by plotting points on the plane. In a few days we will use our graphing calculator to graph equations. The graphing calculator plots points and then "connects the dots". Usually this gives reasonably 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. y = |x| + 2 x −3 −2 y −1 0 1 2 3 y = |x + 2| x −4 −3 y −2 −1 2 3 x x x -2.5 -2 -1.155 2 −1 1 y y y = x −4 x −3 −2 y 0 0 1 2 y 3 y = x3 −4x -5.625 0 3.08 * turning point relative maximum -1 0 1 1.155 3 0 -3 -3.08 2 2.5 0 5.625 * turning point relative minimum y x x Page 2 INTRODUCTION TO GRAPHS AND FUNCTIONS Introductory Topics in Graphing: Symmetry The graphs of y = |x| +2 and y = x2 − 4 exhibit symmetry about the y -axis The graph of y = x3 −4x exhibits symmetry about the origin The graph of x = |y | − 3 exhibits symmetry about the x-axis See page 18 in textbook for more examples of these types of symmetry. Symmetry with respect to x-axis Symmetry with respect to y-axis Symmetry with respect to origin Graphical Tests for Symmetry (page 18) whenever the point (x,y) is on the graph, the point (x, −y) is also on the graph whenever the point (x,y) is on the graph, the point (−x , y) is also on the graph whenever the point (x,y) is on the graph, the point (−x, −y) is also on the graph Algebraic Tests for Symmetry (page 19) replacing y with −y yields an equivalent equation replacing x with −x yields an equivalent equation replacing both x with −x and y with −y yields an equivalent equation Use the algebraic test to determine the type of symmetry, if any, that applies to each of the following: (without looking at the graph) Test for y = x4 − 3x2 symmetry about x axis: replace y by −y y = x5 + 2x y2 = x y = x5 + 2x4 about y axis : replace x by −x about origin replace x by −x and replace y by −y Page 3 INTRODUCTION TO FUNCTIONS Each of these examples 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 example, describe how the numbers in the input set are related to the numbers in the output set. Example 1. Input x 1 2 3 Output y Example 2. Input x 0 1 2 3 4 Describe how the output numbers are related to the input numbers by the arrows Example 3. Input x −1 Output y Output y 1 1 2 2 3 3 4 4 Describe how the output numbers are related to the input numbers by the arrows Example 4. Input x Output y −2 0 0 1 2 2 4 −2 −1 0 1 2 Describe how the output numbers are related to the input numbers by the arrows 0 1 2 Describe how the output numbers are related to the input numbers by the arrows Suppose you had a TRANSFORMATION MACHINE. When you put an input number in, it is transformed into an output number. In each example above, tell whether you can reliably predict what number will come out when you put an input number into the transformation machine. IN OUT The examples where each input can have exactly one output are called functions. Examples 3 and 4 are functions. When you input a number, you can reliably predict which number will be the output. Page 4 Example 4: Absolute Value Function −2 OUT 2 When −2 is input, we know that 2 will be its output. In examples 1 & 2, each input has more than one possibility for its output. When you input a number, you can not predict with certainty which number be the output. Examples 1 & 2 are relations, but they are not functions. Example 2: Less Than Relation (NOT a function) 4 When we input "4", we don't know whether the output will be 1, 2, or 3. OUT ??? The definition of a function in our textbook page 40: A function f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element in the set B. Definition of function in everyday language: Every input has exactly one output If any 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) = (x,y) Look back at the tables for each example and write the ordered pairs below. Explain how you can see from the ordered pairs which is a function and which is not. Example 1: Example 2: Example 3: Example 4: Ways in which functions can be represented: Some functions can be represented in all these ways. Some functions can only be represented in some of these ways. Mathematical description: Multiply the input by 2 to obtain the output value Formula y = 2x Ordered pairs (−1, −2), (0,0), (1, 2), (2,4) Table x: input −1 0 1 2 As sets of ordered pairs As a table As a formula As a graph As a description (sentence form) y: output −2 0 2 4 Verbal description, modeling a real life situation: De Anza College recommends that students study 2 hours outside the classroom for every hour spent in class. x = input = number of hours spent in class y = output = number of hours of study needed outside of class. (Note that x = −1 would not be valid input for this situation) Graph for the given values of x from the table Graph , if the domain includes all real numbers Page 5 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 exactly one number that is its reliable, consistent, predictable output. Identifying a Function from a Formula: Step 1: Solve the equation to put it in y= form: Step 2: If each x value will give only one y value, then it is a function If some x value(s) will give more than one y value, then it is NOT a function x2 − y = 1 x+y=1 −x + y2 = 1 x2 − y2 = 1 Identifying Functions from Graphs Vertical Line Test – Page 55 A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersencts the graph at more than one point. A graph represents a function if whenever you draw a vertical line through any value in the domain, it intersects the graph only once. If any vertical line crosses the graph more than once, it is not a function y y y x A. B. C. x E. y y y x F. D. y x x G. H. Which of the graphs represent functions? PRACTICE: For each graph that represents a function, identify 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 Working with Function Notation Function notation: shows the input and the output: f(s) = s2 Freida’s Fudge Factory 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 x inches. Write the function notation that describes the situation. Substitute into the formula and simplify or evaluate. a. The volume of a square piece of fudge with side of length 2 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 2s inches f. The volume of a square piece of fudge with side of length s + 2 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 12 more pieces of fudge each with side of length 2 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 (1/2)(4πr2) = 2πr2 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) =πr2/200 Write the function notation that describes the situation. Substitute into the formula and simplify or evaluate. a. The amount of paint needed to paint De Anza’s planetarium dome (outdoor radius is approx. 30 ft.) b. The amount of paint needed to paint 2 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 2r 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 2 domes, one with radius 30 feet and another with radius 40 feet. Page 7 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. Example 0: f(x) = x2 + 3x − 2 Domain :_________________________________ Example 1: f(x) = 1 x Domain :_________________________________ Example 2: f(x) = x Domain :_________________________________ Example 3: f(x) = x − 2 Domain :_________________________________ Example 4: f(x) = x+3 Example 5: f(x) = 12 − x Domain : _________________________________ Example 6: f(x) = 20 + x Domain : _________________________________ Domain :_________________________________ Sometimes a function has more than one restriction that affects its domain 1 Example 7: f(x) = x+3 Domain :_________________________________ Example 8: f(x) = x−2 Domain :_________________________________ x−4 Example 9: f(x) = x−2 Domain :_________________________________ x +1 4 x−2 x−2 = 2 x − 3x − 4 ( x + 1)( x − 4) 4 Example 10: f(x) = 3 x−2 x−2 = 2 x − 3x − 4 ( x + 1)( x − 4) Domain :_________________________________ 3 Example 11: f(x) = Example 12:{x f(x) = log x Domain :_________________________________ Domain :_________________________________ Check for even roots: the radicand (or base if in exponential form) can not be negative for even roots, but can be negative for odd roots. Check for division by 0: Set the denominator = 0 and solve for x. EXCLUDE those values from the domain. Check for log functions: The input into a log function must be positive Page 8 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. Example 1: y = f(x) = 2x Range :_________________________________ Example 2: y = f(x) = x2 Range :_________________________________ Example 3: y = f(x) = x2 + 2 Range :_________________________________ Example 4: y = f(x) = −3 Range : _________________________________ Example 5: y = f(x) = Range : _________________________________ x Example 6: y = f(x) = sin x Range : _________________________________ Symmetry of Functions: Even and Odd Functions Graphical Algebraic A function is an even function if: its graph is symmetric f( −x) = f(x) about the y axis whenever x is replaced by −x A function is an odd function if: its graph is symmetric about the origin f( −x) = −f(x) whenever x is replaced by −x Note that if a graph is symmetric about the x axis, 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? y = (x+2)2 y = |x| +2 x = |y | − 3 y = x2 − 4 y = x3 −4x y 45 40 35 30 25 20 15 10 5 0 -4 -3 -2 -1 -5 -10 0 1 2 3 4 5 x -15 20 You should be able to determine symmetry by looking at the graph AND also be able to show it algebraically without looking at the graph. 4. Algebraically, determine whether each of the following functions is even, odd ,or neither. y = g(x) = x5 + 2x y = h(x) = x5 + 2x4 y = f(x) = x4 − 3x2 Page 9 INTRODUCTION TO GRAPHS AND FUNCTIONS: Describing the Behavior of a Function Where a function is shaped like ∪ or part of a ∪, (or a smile) it is concave up. f is increasing if f(a) < f(b) whenver a < b f is decreasing if f(a) > f(b) whenver a < b Where a function is shaped like ∩, or part of a ∩ (or a frown) it is concave down. f is constant if f(a) = f(b) for all a and b A function has a relative maximum y = f(a) at the value x = a when there is no point nearby x = a that has a larger y value. On a graph, this generally is the top of a hill. A function has a relative minimum y = f(a) at the value x = a when there is no point nearby x = a that has a smaller y value. On a graph, this generally is the bottom of a valley. (RULE: an endpoint can't be a relative minimum or a maximum) When a question asks "on what intervals" you need to answer with the appropriate x values, taking into account the fact that x is continuous and is not restricted to integer values. y y x For the function in the graph at the right, on what intervals for x is the function: y y x x x y Increasing? Decreasing? Constant? x Concave up? Concave down? Page 10 INTRODUCTION TO GRAPHS AND FUNCTIONS Piecewise Functions Different equations (formulas) are needed to represent different parts of the function Absolute Value Function y = f(x) = |x| f(x) = g(x) = −x x < 0 x x≥0 −4 x < −1 x−2 −1 ≤ x < 2 (1/2)x x≥2 Step Functions It costs $.25 (25 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 any part of the first hour, and $2 for each additional hour or any part of the additional hour. Draw the graph: The "greatest integer" function: the largest integer that is less than or equal to x y = f(x) = int(x) = x int(2) = int(2.6) = int(π) = int(0) = int(−3) = int(−5.4) = Page 11 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 you can trace the graph of the function without picking up your pencil from the paper; the graph has no gaps or jumps. The graph itself may be a line or a curve. Example 1: The forest service introduced 200 fish into a lake. At the end of 6 years, there are 650 fish. Below are graphs of 3 possible models for how the fish population grew from 200 to 650 fish. a. For the graphs of f and g draw a smooth curve through the points without lifting your pencil. Compare the three graphs. Describe in words how the fish population grows over time in each graph. F 700 I 600 S H 500 F 700 I 600 S H 500 400 300 200 100 F 700 I 600 S H 500 400 300 200 100 0 1 2 3 4 5 6 years 400 300 200 100 0 1 2 3 4 5 6 years 2 L(x) =75x+ 200 0 1 2 3 4 5 6 years 2 g(x) = −12.5x + 150x + 200 f(x) = 12.5x + 200 b. On each graph, draw a line that connects the point where x = 0 to the point where x = 2. Because L(x) is a line, you have traced part of the line, between x = 0 and x = 2. Because f(x) and g(x) are curves, you 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 (x,y) coordinates for x = 0 and x = 2, 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. y − y1 The slope of the secant line is 2 , where (x1, y1) and (x2, y2) are x 2 − x1 the endpoints of the secant line on the graph of the function y = f(x). We usually use function notation f(x) instead of y: f(x 2 ) − f(x1 ) average rate of change = x 2 − x1 If the function is represented by a different letter, such as g, use that letter instead of f. Page 12 f(x 2 ) − f(x1 ) = Average Rate of Change of a function f(x) from x = x1 to x = x2 x 2 − x1 Example 1: (continued) Draw the curves for functions f and g. d For the time period from x = 2 to x = 6 years, draw the secant lines and calculate the average rate of change. F 700 I 600 S H 500 F 700 I 600 S H 500 400 300 200 100 F 700 I 600 S H 500 400 300 200 100 0 1 2 3 4 5 6 years 400 300 200 100 0 1 2 3 4 5 6 years 0 1 2 3 2 L(x) =75x+ 200 g(x) = −12.5x + 150x + 200 f(x) = 12.5x + 200 L: 4 5 6 years 2 f: g: __________________________________________________________________________________________ 40 y Example 2: Evaluate y=f(x) y = f(x) = xx 4x 9 x 3 13x 2 36x using a table on your calculator 30 and fill in below 20 x y x y 10 0 5 0 -2 1 7 2 8 4 9 0 -1 1 2 3 4 6 5 -1 0 7 9 8 x -2 0 -3 0 -4 0 -5 0 Draw the secant lines and calculate the average rate of change (a) from x = 0 to x = 1 (b) from x = 2 to x = 4 (c) from x = 5 to x = 7 (d) from x = 7 to x =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 INTRODUCTION TO GRAPHS AND FUNCTIONS: Difference Quotient and Average Rate of Change Suppose we are at the point (x , f(x)) on a graph and want to move to a nearby point. This nearby point is “h” units away from x along the x axis. Its x-coordinate will be x + h , so its y-coordinate will be y = f(x+h)). If we consider our original point (x,f(x)) as (x1, y1) and our new point (x+h , f(x+h)) as (x2, y2), then the average rate of change formula becomes average rate of change = f(x 2 ) − f(x1 ) f(x + h) − f(x) f(x + h) − f(x) = = x 2 − x1 (x + h) − x h This formula is also called a difference quotient. difference quotient = average rate of change = f(x + h) − f(x) h Example: 2 For the function y = f(x) = x − 7, find and simplify the difference quotient formula. Then evaluate the formula for x = 6 and h = 0.5 Page 14 Library of Functions Graph the following equations by hand or using your graphing calculator. If using the calculator, use the standard graphing window: −10 ≤ x ≤ 10, −10 ≤ y ≤ 10 ; xScl and yScl = 1 TI-86 GRAPH F3:ZOOM F4:ZSTD TI−82, 83, 84 ZOOM 6:Standard TI:83&84 Find abs and int on the MATH NUM menu TI:86 Find abs and int on the 2nd Math F1:NUM menu Sketch the graph of each functions on the worksheet below. Save this for reference. Study, learn & remember the shapes of these different basic types of functions. Absolute Value Function Linear Functions & Equations y = | x | = abs(x) y=c y=x y=−x x=c Constant Function Identity Function (not a function) Even Integer Powers of x y=x y = x6 Quadratic Function 2 1/2 y=x Even Roots of x = x Square Root Function Reciprocal Function 1 y = x −1 = x 3 y=x Cubic Function Odd integer powers of x y = x5 Odd Roots of x 1/4 y=x = 4 x 1/3 y=x = 3 x y = x1/5 = 5 x Cube Root Function More Negative Powers of x 1 1 y = x −2 = 2 y = x −3 = 3 x x Step Function y = x = int(x) Page 15 Transformation of Functions (Refer also to Transformations Homework Worksheet) To find the graph of f(x) −c, c > 0, f(x) +c, c > 0 Describe the transformation of the graph of y = f(x) f(x−c), c > 0, f(x+c), c > 0 cf(x), 0 < c < 1 cf(x), c > 1 f(cx), 0 < c < 1 f(cx), c > 1 −f(x) f(−x) When doing more than one transformation, order matters. Not following the correct order may give incorrect results Vertical Transformations: Reflections and stretches/compressions have first priority and can be done in any order. Do vertical shifts up or down after vertical reflections, stretches or compressions. Horizontal Transformations: (2x − 6) = (2(x−3)) Factor the expression inside the function. Examples: (cx−d) = c x − d c First shift function horizontally by d/c units (right if d/c > 0, left if d/c < 0) Next do horizontal stretches/compressions using the line x = d/c as the axis of reference Then do horizontal reflections using the line x = d/c as the axis of reference For each problem the graph of y = f(x) is shown. Describe the transformations of the graph of y = f(x) to obtain the given function. Then use the transformations to draw the graph of the given function. g(x) = 3f(x+2) h(x) = 3f(–4x) s(x)= .5f(.5x) –3 Describe the transformations of the graph of y = |x| to obtain the given function. Then use the transformations to draw the graph of the given function. y = u(x) = –|x|+3 y = v(x) = |x+3| –2 y = w(x) = |2x–6| Page 16 Transformations of Functions: Finding Equations From Graphs A B y 3 2 -3 -1 -2 10 0 1 2 x 3 -1 -2 (-1,-4) -3 -4 (1,-4) -5 -6 (-2,-7) -7 (2,-7) -8 -9 -10 Equation________________________ -11 Equation______________________ -12 C D Equation________________________ E Equation________________________ Equation______________________ F Equation______________________ Page 17 INTRODUCTION TO GRAPHS AND FUNCTIONS: Algebraic Combination of Functions Functions can be added, subtracted or multiplied A function can be divided by 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 x that make the denominator equal to 0 must be removed from the domain. x For the functions f(x) = and g(x) = x − 2, find the sums, differences and products, and quotients: Arithmetic Combination of Functions Domain ( f+g ) (x) = f(x) + g(x) = ( f−g ) (x) = f(x) − g(x) = ( fg ) (x) = f(x)g(x) = f ( x ) g g ( x ) f 2 For the functions f(x) = x −3x+2 and g(x) = x − 2, find the combinations requested: Arithmetic Combination of Functions ( g−f ) (x) = g(x) − f(x) = Domain ( fg ) (x) = f(x)g(x) = f ( x ) g g ( x ) f For the functions f and g shown in the graph, find (f+g)(2) (f−g) (2) (fg) (−3) f (3) g f (1) g g (1) f y g f x Page 18 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!: Always work "from the inside to the outside" or "from right to left" 2 f(x) = x g(x) = x + 1 Find f°g (x) = f(g(x)) Find g°f (x) = g(f(x)) Evaluate f°g (−2 ) Evaluate g°f (−2 ) Composition is NOT the same as multiplication; they usually do not give the same results Find the product (fg) (x) = f(x)g(x) The product obtained by multiplying the functions is not the same as the compositions above. f(x) = 2x + 1 2 g(x) = 9 − x f°g (x) = f(g(x)) g°f (x) = g(f(x)) Application 1 You are going to buy a desk at an office supply store. You have 2 coupons, one for $25 rebate for a purchase of any desk; the other for 20% of your entire purchase. a. If only the $25 rebate is given, express the sale price y as a function of x: y = R(x) = _________ If only the 20% discount is given, express the sale price y as a function of x: y = D(x) = ________ b. Suppose the store is willing to give both the rebate and the discount. You want to know if it is better for you, the consumer, if the store calculates the 20% discount first or the $25 rebate first. Assume the desk costs $200. Find cost for each situation: $25 Rebate Only: 20% Discount Only $25 Rebate first, then 20% Discount 20% Discount first, then $25 Rebate c. For each situation below, express the sale price using a composition of functions R(x) and D(x) for a desk with a list price of $x $25 Rebate first, then 20%Discount y = ____________________________________________ 20%Discount first, then $25 Rebate: y = ______________________________________________ Page 19 COMPOSITION OF FUNCTIONS Application 2 You are on a boat in Lake Tahoe and accidentally drop your sunglasses into the lake. At the spot where your glasses hit the water, a small wave forms, and concentric circular waves begin to propagate out from the spot where your 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(x). Specify f and g. BE CAREFUL ABOUT THE ORDER! You must choose functions f, g, so that f(x)≠x, g(x) ≠x U(x) = T(x) = cos 3 x x +8 W(x) = (x2 + 3x +2)6 Z(x) = 2 x+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(x) ≠ x, g(x) ≠ x, h(x) ≠ x M(x) = tan ( x3 + 4 ) P(x) = 1 (2 x + 1)3 C(x) = cos3(2x+1) D(x) = 1− sin x Distinguishing Products and Compositions: A product of functions is obtained when we multiply two functions by 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(x) = x sin x b. Q(x) = 10 4 x −1 c. S (x) = cos (sin x) x d. U(x) =3xe J(x) = sin (x2) R(x) = (4x − 1)10 x T (x) = cos x sin x 3x V(x) = e Page 20 COMPOSITION OF FUNCTIONS: Graphical Interpretation y Using the functions given in the graph, find (f° g) (−1) g f (g° f) (2) x (f° g) (3) (yes it’s a “trick question”) COMPOSITION OF FUNCTIONS: Domain To find the domain of a composition of functions (f°g): (f° g)(x) = f(g(x)) use this 3 step process: Step 1: Find the domain (set of legal input) for g Step 2: Then find the outputs from g that are not legal inputs to f Find the value(s) of x that is/are input to g to create that output Step 3: Eliminate those values of x from the domain of g to get the domain of (f° g) When you compose two functions, it is important to specify the domain of the composition, using this method. Warning: If you just find the formula for f(g(x)) and do not use this process to find the domain, you may get in incorrect answer for the domain of f(g(x)) Example A: Example 5 in text : f(x) x2 − 9 g(x) = 9 − x 2 : Find the domain of f(g(x)) Step 1: Domain of g is –3 ≤ x ≤ 3 Step 2: All output values from g are legal input to f Step 3: The domain of f(g(x)) is –3 ≤ x ≤ 3 [–3 , 3 ] If you just found the formula for f(g(x)) = − x2 you would incorrectly think the domain is all real numbers 1 and g(x) = x − 100 : Find the domain of f(g(x)) x2 Step 1: Domain of g is x ≥ 100 Example B: f(x) = Step 2: BUT when g has output g(x) = 0, then you can't find f(x) because of division by 0 The output g(x) = 0 occurs when x = 100 Step 3: Therefore x = 100 must be eliminated from the domain The domain of f(g(x)) is x > 100 (100 , ∞ ) 1 x − 100 you would incorrectly think the domain is all real numbers except 100 If you just found the formula for f(g(x)) = Page 21 INTRODUCTION TO GRAPHS AND FUNCTIONS: One to One Functions A function is ONE TO ONE if there is one x value for every y value. You can trace any y value back to its corresponding x value, and no other y value has the same x. This corresponds to a HORIZONTAL LINE TEST A graph represents a ONE TO ONE FUNCTION if whenever you draw a HORIZONTAL line through any value in the domain, it intersects the graph only once. If any horizontal line crosses the graph more than once, it is NOT a ONE TO ONE function Note that it must also satisfy the vertical line test to be a function: A graph represents a function if whenever you draw a vertical line through any value in the domain, it intersects the graph only once. If any vertical line crosses the graph more than once, it is not a function y y x x y = f(x) y y y = g(x) y = s(x) y = u(x) x y = w(x) INTRODUCTION TO GRAPHS AND FUNCTIONS: Inverse Functions An inverse f −1 Intuitive Definition of Inverse Function of a function f "undoes" f, bringing you back to the original input Condition For Existence 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 y values that are output from more than one x-value as input.) Which functions shown above have inverse functions?_________________ Which functions shown above do NOT have inverse functions?_________________ Give a numerical example that explains why y = f(x) = x2 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 Example: −1 °f (x) = x AND f°f −1 (x) = x x−7 12 Show that these functions are inverses of each other using composition: (SHOW YOUR WORK!) f(x) = 12x + 7 1 1 f − (x) = 1 f − °f (x) = f− (f (x)) = 1 1 f °f − (x) = f (f − (x)) = Page 22 INTRODUCTION TO GRAPHS AND FUNCTIONS: Inverse Functions y Graphical Example: 1 g 1 Find g− (3) Find g − (2) 1 x 1 Find (g− )(-1) Find g− (0) Ordered Pairs: If you represent a function as ordered pairs, the roles of x and y are switched when you write the inverse function as ordered pairs. f(x) = x 1 3 f − (x) = x {(− 3, −27), (−2, −8), (−1, −1),(0,0), (1,1), (2,8), (3,27)} 1/3 = 3 x {__________________________________________________} Domain and Range: The domain of the function y = f(x) becomes the range of the inverse y = f −1 (x) The range of the function y = f(x) becomes the domain of its inverse y = f −1 (x) Graphical Symmetry Graph f(x) = x 3 1 and f − (x) = x 1/3 =3x Accurately plot the points for x = −2, −1, 0, 1, 2 and connect them with a smooth curve to represent the functions. Use different colors. Draw the line y = x. 1 The graph of f − is the reflection of the graph of f across the line y = x. Draw the line y = x and use it as a guide to help draw the graph of g−1 Domain of g: _______________ 1 Domain of g− : _______________ Range of g: _______________ y g x 1 Range of g− : _______________ Page 23 INTRODUCTION TO GRAPHS AND FUNCTIONS: Inverse Functions Finding the inverse function algebraically: 1. First determine whether f has an inverse (Is f one-to-one? Horizontal Line Test) 2. In the equation for f(x), replace f(x) by y 3. Solve for x = This is NOT the same order as the instructions in the textbook, but will lead to the same result. 4. Interchange the letters x and y 1 5. Replace y by f − (x) in the new equation 1 6. Verify that f and f− are inverses using composition Examples: Each of the following functions is one-to-one and has an inverse. Find the inverse function for each. 5 − 3x x+3 A. f(x) = (p. 97) D. g(x) = (p. 99, #24) 2 x−2 B. f(x) = 12x + 7 C. f(x) = 3 x +1 E. P(x) = 75 x , 100 − x (p. 98) Page 24

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