Chapter Analysis of Graphs of Functions Chapter Analysis of Graphs of Functions Covered in this Chapter:.1 Graphs of Basic Functions and their Domain and Range. Odd, Even Functions, and their Symmetry.. Translations of Graphs: Vertical and Horizontal Shift, Stretching and Compressing, and Reflections.. Graphs of Absolute Value Functions. Equations, Inequalities, and Applications..4 Piecewise Functions, Greatest Integer Function, and their Graphs..5 Algebra of Functions and their Composition. Difference Quotient. Answers to Eercises. -48-
Chapter Analysis of Graphs of Functions Definitions: Continuous Functions:.1 Graphs of Functions A function is called continuous on its domain if it has no tears, gaps, or holes. That is, if it can be drawn without lifting your pencil when you draw it on paper. Increasing and Decreasing Functions: A function f is called increasing on an interval I if f f whenever in I 1 1 A function f is called decreasing on an interval I if f f whenever in I 1 1 y B D y f C A f f 1 0 a 1 b c d Increasing Increasing Decreasing Constant Functions: A function f is called constant on an interval I if f f whenever in I (or in general, whenever ) 1 1 1 10 8 6 4 y y 5 Symmetry of Functions: Even Functions 10 8 6 4 0 4 6 8 10 y f y A function f is called an EVEN function if it satisfies the condition f f for all in its domain. i.e. If the point, is on the graph of, then the point, must be also on its graph. An EVEN function is symmetric about the y-ais. 4 6 8 10-49-
Section.1 Graphs of Functions Eamples of Even Functions y f f 0 Odd Functions y f y A function f is called an ODD function if it satisfies the condition f f for all in its domain. i.e. If the point, is on the graph of, then the point, must be also on its graph. An ODD function is symmetric about the origin. Eamples of Odd Functions y 0 f f Tests for Symmetry: Symmetry about the y-ais (even functions): Replace with in f ( ). If f ( ) f ( ), then the graph is symmetric about the y-ais. Symmetry about the origin (odd functions): Replace with in f ( ). If f ( ) f ( ), then the graph is symmetric about the origin. Symmetry about the -ais: Replace y with y in the equation. If you get the same equation, then the graph is symmetric about the -ais. Remember that graphs symmetric about the -ais are not functions. See the eample below. -50-
Chapter Analysis of Graphs of Functions Eample 1: Decide whether the function is odd, even or neither. 4 Solution: First replace with in the given function. 4 f 5 10 f 4 ( ) ( ) 5( ) 10 f f 4 ( ) 5 10 ( ) f ( ) f ( ) The given function is EVEN. (i.e. The function is symmetric about the y-ais.) f 5 10 Eample : Decide whether the function is odd, even or neither. Solution: First replace with in the given function. f 5 4 f ( ) 5( ) 4( ) f ( ) 5 4 f ( ) (5 4 ) f ( ) f ( ) nor f ( ) The given function is NEITHER odd nor even. f 5 4 Eample : Decide whether the function is odd, even or neither. Solution: First replace with in the given function. 5 f 4 1 ( ) 5( ) f 4( ) 1 f f 5 4 1 5 4 1 f ( ) f ( ) The given function is ODD. (i.e. The function is symmetric about the origin.) f 5 4 1-51-
Basic Functions and their Graphs Section.1 Graphs of Functions Identity Function Absolute Value Function y Domain: or (, ) Range: or (, ) Odd Function Symmetric about the origin y Domain: or (, ) Range: y 0 or [0, ) Even Function Symmetric about the y-ais Quadratic Function Square Root Function y Domain: or (, ) Range: y 0 or [0, ) Even Function Symmetric about the y-ais y Domain: 0 or [0, ) Range: y 0 or [0, ) Function is neither odd nor even Cubic Function y Domain: or (, ) Range: or (, ) Odd Function Symmetric about the origin -5-
Chapter Analysis of Graphs of Functions Cube Root Function y Domain: or (, ) Range: or (, ) Odd function Symmetric about the origin ******************************************************** Eercises.1 In eercises 1 1, decide whether the function is odd, even, or neither and state whether its graph is symmetric about the origin, y-ais, or neither. 4 1) f ) f 4 ) f 5 4) f 4 1 5 5) f 6 6) f 4 5 1 7) f 1 8) f 4 9) f 1 10) f 4 5 5 11) f 4 5 1 5 1) f 4 5-5-
Section.1 Graphs of Functions In eercises 1 4, decide whether the graph is symmetric about the -ais, y-ais, or the origin. 1) 14) 15) 16) 17) 18) 19) 0) 1) ) ) 4) -54-
Chapter Analysis of Graphs of Functions. Translations of Graphs In this section, translations of graphs will be discussed. We will be using a screen produced by a TI 84+ showing an original function drawn in a thick line and some other functions that were produced from that original function. We should be able to come up with a conclusion from all of these screens. Vertical Shifts The above two screens show that if f ( ) is a function and c is a positive number, then the graph of f ( ) c is eactly the same as the graph of f ( ) but shifted down c units. The above two screens show that if f ( ) is a function and c is a positive number, then the graph of f ( ) c is eactly the same as the graph of f ( ) but shifted up c units. Horizontal Shifts The above two screens show that if f ( ) is a function and c is a positive number, then the graph of f ( c) is eactly the same as the graph of f ( ) but shifted to the right c units. -55-
Section. Translations of Graphs The above two screens show that if f ( ) is a function and c is a positive number, then the graph of f ( c) is eactly the same as the graph of f ( ) but shifted to the left c units. Vertical Stretching The above screens show that if f ( ) is a function and c is a positive number greater than 1 ( c 1), then the graph of [ cf ( )] is eactly the same as the graph of f ( ) but vertically stretched by a factor of c Vertical Shrinking The above screens show that if f ( ) is a function and c is a positive number such that 0 c 1, then the graph of [ cf ( )] is eactly the same as the graph of f ( ) but vertically shrunk by a factor of c. -56-
Chapter Analysis of Graphs of Functions Horizontal Shrinking The above screens show that if f ( ) is a function and c is a positive number greater than 1 ( c 1), then the graph of f ( c) is eactly the same as the graph of f ( ) but horizontally shrunk by a factor of c. Horizontal Stretching The above screens show that if f ( ) is a function and c is a positive number such that 0 c 1, then the graph of f ( c) is eactly the same as the graph of f ( ) but horizontally stretched by a factor of c. Reflection About the -ais The above screens show that if f ( ) is a function, then the graph of [ f ( )] is eactly the same as the graph of f ( ) but reflected about the -ais. -57-
Section. Translations of Graphs Reflection About the y-ais The above screens show that if f ( ) is a function, then the graph of f ( ) is eactly the same as the graph of f ( ) but reflected about the y-ais. Now, that we know all operations that can be performed on a given graph, we should be able to answer the following eamples. Eample 1: The graph of the function f ( ) is shifted to the right units, then vertically stretched by a factor of 4, then reflected about the y-ais, and finally shifted down 5 units. Write the final translated function h() and give its graph. Solution: f ( ) Shifted to the right units Vertically stretched by a factor of 4 4 Reflected about y the -ais 4 Shifted down 5 units 4 5 h( ) -58-
Chapter Analysis of Graphs of Functions Eample : The graph of the function f ( ) is shifted to the left 4 units, then vertically shrunk by a factor of 1/, then reflected about the -ais, and finally shifted up units. Write the final translated function h() and give its graph. Solution: f ( ) Shifted to the left 4 units 4 Vertically shrunk by 1 a factor of 1/ 4 Reflected about 1 the -ais 4 Shifted up units 1 4 ( ) h Eample : The graph of the function on the graph of the function f is obtained by performing the following transformations g in the given order: shifted horizontally 5 units to the left, then vertically stretched by a factor of 4, reflected about the y-ais, and finally shifted downward units. Find the rule of the function g. Solution: We will show two methods to solve these types of problems. Method 1: Following the eact steps as given and equating the final answer to -59- f. shifted to the left vertically stretched reflected about 5 4 5 4 5 g g g g 5 units by a factor of 4 the y-ais shifted down 4g 5 4g 5 f ; solve for g units 4g 5 ; add to both sides of the equation 4g 5 ; multiply both sides of the equation by 1/4
1 g 5 ; reflect about the y-ais 4 1 1 g g 4 4 5 ; shift to the right 5 units to obtain g 1 5 4 Section. Translations of Graphs Note: The absolute value of any negative number is equal to the absolute value of the number itself. Method : The function g is shifted to the left 5 units, vertically stretched by a factor of 4, reflected about the y-ais, and finally shifted down units to obtain the function Perform the opposite of each transformation in reverse order on g. The function f f. to obtain the rule of the function f is shifted up units, reflected about the y-ais, vertically shrunk by a factor of 1/4, and finally shifted to the right 5 units. Shift Reflect f up units: f f about the y-ais: f f f 1 Vertically shrink f Shift 1 f 4 1 g 5 4 1 1 by a factor of 1 4 : f f 1 1 4 4 to the right 5 units: f 5 5 g 1 4-60-
Chapter Analysis of Graphs of Functions Eercises. In eercises 1 8, write the function g whose graph can be obtained from the graph of the function f by performing the transformations in the given order. Graph the function g. 1) f ; shift the graph horizontally units to the right and then vertically downward 5 units. ) f 1; shift the graph horizontally units to the left and then vertically upward 4 units. ) f 1; reflect the graph about the y-ais, then shift it vertically downward units. 4) f 4 4; reflect the graph about the y-ais, then shift it vertically downward 1 unit. 5) f 1; reflect the graph about the -ais, then shift it vertically upward units. 6) f 4 4; reflect the graph about the -ais, then shift it vertically upward units. 7) f ; shift the graph horizontally to the right 4 units, stretch it vertically by a factor of, reflect the graph about the y-ais, then shift it vertically upward 5 units. 8) f ; shift the graph horizontally to the left units, stretch it vertically by a factor of, reflect the graph about the y-ais, then shift it vertically downward 1 unit. In eercises 9 14, use the graph of the function g in the given figure to sketch the graph of the function f. 9) f g 10) f g 11) f g 1) f 0.5g 1) f g 14) f g -61-
Section. Translations of Graphs In eercises 15 0, use the graph of the function g in the given figure to sketch the graph of the function f. 15) f g 1 16) f g 17) f g 18) f 0.5g 19) f g 0) f g 1 1) The graph of the function f is obtained by performing the following transformations on the graph of the function g in the given order: shifted horizontally units to the right, then vertically shrunk by a factor of 0.5, reflected about the -ais, and finally shifted downward units. Find the rule of the function g. ) The graph of the function f is obtained by performing the following transformations on the graph of the function g in the given order: shifted horizontally units to the left, then vertically stretched by a factor of, reflected about the y-ais, and finally shifted upward 5 units. Find the rule of the function g. ) The graph of the function f is obtained by performing the following transformations on the graph of the function g in the given order: shifted horizontally 1 unit to the right, then vertically shrunk by a factor of 0.5, reflected about the y-ais, and finally shifted downward units. Find the rule of the function g. 4) The graph of the function f is obtained by performing the following transformations on the graph of the function g in the given order: shifted horizontally 4 units to the left, then vertically shrunk by a factor of 0.5, reflected about the -ais, and finally shifted downward units. Find the rule of the function g. -6-
Chapter Analysis of Graphs of Functions. Absolute Equalities and Inequalities Absolute Value of a Function: f f if f 0 f if f 0 The above notation is simply saying that when the graph is zero or positive, leave it as it is. If the graph is negative then reflect it about the -ais to make it positive. 4 Eample 1: Given that f 0.1, use your graphing calculator to graph y f. Solution: The first screen below shows the graph of the function f(). The second screen shows both the graph of the function in thick lines and the graph of its absolute together. The last screen shows only the graph of the absolute of the function. Absolute Equations: if 0 if 0 Absolute value of measures the distance from to zero. Remember that distance must be non-negative. Therefore 7 7, 1 1, 0 0, 5 5, 10 10, and so on since the distance from any number to zero is the positive case of that number. Eample : Solve the absolute equation. Solution: The problem is simply saying that if the distance from to zero is units, what is? We know that the distance must be non-negative. If we want to measure units from zero, we have two ways to do so. Either we count units to the right (this case we get +) or we can count units to the left (this case we get -). This means we have two answers for, + and -. The figure below shows how we got these two answers. units units 1 0 1-6-
Section. Absolute Equations and Inequalities To generalize the method of solving absolute equations, we have the following rules: If a b c where c is a positive number ( c 0) then, a b c or a b c If a b c where c is a negative number ( c 0) then the equation has no solution If a b 0 then, a b 0 Eample : Solve the absolute equation 5. Solution: First we need to apply the rule above and remove the absolute symbol by writing two equations. 5 5 or 5 5 ;, 8 8 Eample 4: Solve the absolute equation 7 6 5. Solution: We have to isolate the absolute epression before removing the absolute symbol. Therefore the equivalent equation is 7 11. Now we should be able to write two equations and solve the problem. 7 11 18 or 7 11 4 ;,9 9 Eample 5: Solve the absolute equation 7 8. Solution: We have to isolate the absolute epression before removing the absolute symbol. Therefore the equivalent equation is 7 5. But, we know that absolute value must be non-negative all the time. Therefore, the given absolute equation has no solution. Eample 6: Solve the absolute equation 10 0. Solution: In this type of problems when you see zero on one side of the equation, simply remove the absolute symbol and solve the equation as usual. 10 0 10 0 10 5-64-
Chapter Analysis of Graphs of Functions Solving Absolute Equations of the Type a b c d If a b c d, then a b c d or a b c d Eample 7: Solve the absolute equation 7 5. Solution: Follow the above rule and write two equations to solve the problem. 7 5 1 1 7 5 7 5 5, 5 1, 5 Eample 8: Solve the absolute equation 5. Solution: Write two equations and solve the problem. 5 5 07 No Solution 5 5 6 1 1 Note: In the last eample, don t write the answer is no solution or -1/. Just write -1/ Solving Inequalities of the Form a b c Solving Absolute Inequalities If a b c where c is a positive number c 0, then a b c and a b c If we write the above as continued inequalities, we can write the inequality as follows: c a b c If a b c where c is a negative number c 0, then the inequality has no solution If a b 0, then a b 0 Note: In the above rule, due to space limitation, we used the symbol of inequality less than or equal. The less than symbol also applies to this rule. -65-
Section. Absolute Equations and Inequalities The figure below shows why we call the above inequality the AND case. Notice that the interval of the solution is one continuous interval. If is in the interval (-, ), we can see that is less than AND is greater than -. less than units less than units 1 0 1 Eample 9: Solve the absolute inequality. 5 Solution: First we need to apply the rule above and remove the absolute symbol by writing continued inequalities as follows: 5 5 5 5 8 Set Notation: { 8 } Interval Notation: [-8, ] Eample 11: Solve the absolute inequality. 7 8 Solution: We have to isolate the absolute epression before removing the absolute symbol. Therefore the equivalent inequality is 7 5. But we know that absolute value must be nonnegative all the time. Therefore, the given absolute inequality has no solution. Eample 10: Solve the absolute inequality. 7 6 5 Solution: We have to isolate the absolute epression before removing the absolute symbol. Therefore the equivalent inequality is 7 11. Now we should be able to write the inequality as continued inequalities and solve the problem. 11 7 11 11 7 7 7 11 7 4 18 9 Set Notation: { 9} Interval Notation: (,9) Eample 1: Solve the absolute inequality. 10 0 Solution: In these types of problems when you see that the inequality epression is less than or equal to zero, remove the absolute symbol, replace the inequality symbol with an equal sign and solve the equation. 10 0 10 0 10 5-66-
Chapter Analysis of Graphs of Functions Solving Inequalities of the Form a b c If If a b c where c is a positive number c 0, then a b c or a b c If a b c where c is a non-positive number c 0, then the solution is the set of all real numbers b a b 0, then the solution is all real numbers ecept a b b that is:,, a a Note: In the above rule, due to space limitation, we used the symbol of inequality greater than or equal also applies to this rule.. The greater than symbol The figure below shows why we call the above inequality the OR case. Notice that the interval of the solution is two separate intervals. If is in the interval, then is less than -. If is in the interval, then is greater than. Since cannot be in two places at the same time, we have to say that is less than - OR is greater than. greater than units 1 0 1 greater than units Eample 1: Solve the absolute inequality 5 Solution: First we need to apply the rule above and remove the absolute symbol by writing the absolute inequality as two separate inequalities. 5 5 8 or 5 5 Set Notation: { 8 or } Interval Notation: (-,-8] [, ) Eample 14: Solve the absolute inequality 7 8 Solution: We have to isolate the absolute epression before removing the absolute symbol. Therefore, the equivalent inequality is 7 5. But, we know that absolute value must be non-negative all the time. Therefore, the solution to the given absolute inequality is the set of all real numbers. -67-
Section. Absolute Equations and Inequalities Eample 15: Solve the absolute inequality 7 6 5 Solution: We have to isolate the absolute epression before removing the absolute symbol. Therefore, the equivalent inequality is 7 11. Now we should be able to write the inequality as two separate inequalities and solve the problem. 7 11 7 7 11 7 4 or 7 11 7 7 11 7 18 9 Set Notation: { or 9} Interval Notation: (, ) (9, ) Eample 16: Solve the absolute inequality 10 0 Solution: In these types of problems when you see that the inequality epression is greater than zero, remove the absolute symbol, replace the inequality symbol with an equal sign and solve the equation. 10 0 10 0 10 5 The solution is all real numbers ecept 5 or ecept 5. Interval notation: (-,5) (5, ). Now that we have discussed all types of absolute equalities and inequalities, we should be able to solve an eample with absolute equality and its related inequalities. Eample 17: Solve the absolute equality and its given related inequalities. Solution: a) 7 b) 7 c) 7 a) 7 7 or 7 7 or 7 4 or 10 or 5 {,5} b) 7 7 7 7 7 4 10 5 Interval Notation: [,5] Set Notation: { 5} -68- c) 7 7 or 7 7 or 7 4 or 10 or 5 Interval Notation: (, ] [5, ) Set Notation: { or 5}
Chapter Analysis of Graphs of Functions Eample 18: Use your graphing calculator to solve the absolute equality and its related inequalities. a) 1 6 10 b) 1 6 10 c) 1 6 10 Solution: First enter the left side of the equality as Y 1 and the right hand side as Y. Move cursor close to the other intersection From the final screen we can see that: a).5,7.5 b) [-.5,7.5] c) (,.5] [7.5, ) -69-
In eercises 1 16, solve the absolute equations. Eercises. Section. Absolute Equations and Inequalities 1) 5 4 ) ) 5 4) 1 7 5) 7 8 6) 4 1 7) 5 6 10 8) 4 5 4 9) 5 10) 5 1 9 5 11) 6 5 5 1) 9 7 7 1) 5 14) 5 5 15) 1 16) 5 4 5 1 In eercises 17 40, solve the absolute inequalities. 17) 4 18) 1 8 19) 5 0) 4 1 1) 4 5 ) 4 6 ) 10 5 5 4) 6 5) 5 6 6) 6 5 7 7) 10 11 8) 6 4 4 9) 5 0) 4 1) 5 ) 5 4 1 ) 4 4) 5 8 5) 10 11 11 6) 6 7 7 7) 5 6 8) 6 5 7 9) 10 9 9 40) 6 5 5 In eercises 41 46, solve the absolute equation and its related inequalities (see eample #17). 41. a) 1 4. a) 6 4. a) 1 9 b) 1 b) 6 b) 1 9 c) 1 c) 6 c) 1 9 44. a) 4 5 45. a) 5 11 46. a) 4 5 b) 4 5 b) 5 11 b) 4 5 c) 4 5 c) 5 11 c) 4 5-70-
Chapter Analysis of Graphs of Functions.4 Piecewise Functions Definition: A piecewise function is a defined function that consists of several functions (i.e. The graph of a piecewise function consists of several graphs. Each one of these graphs is also a function). Graphs of parts of a piecewise function usually have one or more of the following: holes, sharp corners, and jumps. Eample 1: Graph the following piecewise function and evaluate it at the given values., if 0 f ; f, f, f 0 5, if 0 Solution: To graph the given piecewise function, we need to graph each part separately. Be sure to graph each part within its defined domain. You can see that the first part which is a parabola is defined only over the interval (,0] and the point zero is included (closed circle). The second part represents a line. The line is defined only on the interval (0, ) where point zero is ecluded (open circle). Now, press TRACE and enter the values for at which you want to evaluate the function f. f f f 5 1, use the second function since is in its domain only. 4, use the first function since - is in its domain only 0 0 0, use the first function since 0 is in its domain only Computer generated graph : -71-
Section.4 Piecewise Functions Note: There are other ways to enter the piecewise function of eample #1 in our TI. See the following screens to see how this can be done. In our opinion, the first method is easier than these other methods. Method Method Greatest Integer Function (Step Function): Notation: f ; which means the greatest integer that is less than or equal to. Equality will occur only when is an integer. If not, always round down. Eample : Evaluate the following: f.4, 5, 1., 0, 1.45, 7 Solution: Since the value of the function is equal to if is an integer, we need to round down if is not f, 5, 1,0,, 7 an integer, Graphing Calculator Solution of Eample : Press MATH NUM 5:int( Graph of the Greatest Integer Function f : Computer G enerated G raph : Domain: The set of all real numbers or, Range: The set of all integers -7-
Chapter Analysis of Graphs of Functions Using a graphing calculator to graph f : To find "int", press MATH NUM 5:int( Don t be deceived by the graph on the previous page. This is a lack of technology. All vertical segments are not part of the graph. If you want to see a better graph, you need to change the MODE in your TI from Connected to Dot. To do this, press MODE, move the cursor down to Dot and press ENTER to highlight Dot. If you do this you ll get the screen below. Now you can see why we call this function Step Function. Note: The TI graphing calculator will not show the open or closed end of each segment. Eample : Graph the function f 1, evaluate f 1., f, f., f 4,and f 7.1 Solution: Since the value of the function is equal to if is an integer, round down if is not an integer. f f f f f 1. 1.1. 1 4 4..1 1. 1 4 4 1 7.1 7.11 6.1 6-7-
Section.4 Piecewise Functions Computer G enerated G raph : Using the Table Feature: Press nd TBLSET, highlight Ask to the right of Indpnt:, and highlight Auto to the right of Depend:. Press nd TABLE, enter the value for at which you want to evaluate the function, and then press ENTER. From the last screen, f f f f f 1., 4,. 1, 4, and 7.1 6. -74-
Chapter Analysis of Graphs of Functions Eercises.4 In eercises 1 10, graph the function and evaluate it at the given values. if 1) f, evaluate f 4, f 5,and f 5 if if 1 ) f, evaluate f 4, f 5,and f 1 1 4 if 1 1 if 1 ) f, evaluate f 0, f 1, f 1,and f if 1 if 4) f, evaluate f 0, f, f,and f if if 0 5) f, evaluate f 0, f 1,and f 1 4 if 0 if 0 6) f, evaluate f 0, f 1,and f 1 if 0 if 1 7) f, evaluate f, f 1,and f 5 if 1 if 8) f, evaluate f 0, f,and f if 1 if 9) f if 1, evaluate f 0, f, f 1, f,and f 4 1 if 1 5 if 1 10) f if 1, evaluate f 0, f 1, f, f,and f 4 9 if -75-
Section.4 Piecewise Functions In eercises 11 18, graph the function and evaluate it at the given values. 11) f, evaluate f., f 5, f 4.1, f,and f 5.1 1) f 1, evaluate f 1., f, f., f 4,and f 7.1 1) f 1, evaluate f 1., f 5., f 0, f.5,and f 7 14) f, evaluate f., f 4, f 0.1, f 6,and f 8.1 15) f 1, evaluate f 1., f, f.1, f,and f 4.1 16) f 1, evaluate f 4., f 1, f 0, f 5,and f.1 17) f, evaluate f., f 1, f 0, f,and f 7. 18) f, evaluate f 5., f, f 0, f.,and f 5-76-
Chapter Analysis of Graphs of Functions.5 Algebra of Functions and their Composites Two functions f and g can be combined to form a new function h eactly the same way we add, subtract, multiply, and divide real numbers. Let f and g be two functions with domains A and B respectively then: 1 f f g g h ( ) f g f g domain { A B} h ( ) f g f g Algebra of Functions domain { A B} h ( ) fg f g domain { A B} h4 ( ) domain { A B and g 0} The graph of the function h( ) ( f g)( ) can be obtained from the graph of f and g by adding the corresponding y-coordinates as shown on the figure below. Similarly, we can find the difference, product, and division of f and g. In case of division, remember that g ( ) 0. So any value of that makes the denominator equal to zero must be ecluded. f Eample 1: If f ( ) 5 and g( ), find ( f g)( ), ( f g)( ), ( fg)( ), the ( )( ). g f Also evaluate ( f g)(1) and ( )(). g Solution: f g ( ) f ( ) g( ) 5 f g ( ) f ( ) g( ) 5 5 8 fg ( ) f ( ) g( ) 5 15 f f ( ) 5 ( ), g g( ) f To evaluate f g(1), and () : g f g (1) (1) 1-77- f () 5 1 () g 5
Notation: f g f g Section.5 Algebra of Functions and their Composites Composition of Functions The above can be read in many ways. It can be read as f composed with g at, f composed with g, the composition of f and g, f of g of, or f circle g. The Composite Function f gis defined by f g f g Its domain is all numbers in the domain of g such that g is in the domain of f Eample : If f g f g g f Solution: 5 and, find and. f g f g g 5 5 6 4 g f g f f 5 6 7 Note: In general f g g f Eample : If f 7 and g 5 1, find f g Solution: We can use two different methods to solve the problem. First Method: Find f g f g and then substitute for f g f g g 7 5 1 7 10 5 10() 5 5 Second Method: g f g f g, evaluate g and substitute the result into f 5 1 15 1 16 f g f g f 16 16 7 7 5-78-
Chapter Analysis of Graphs of Functions The Difference Quotient The difference quotient epression is very important in the study of calculus. Looking at the graph below, the difference quotient is the slope of the line PQ which is called the secant line. f y Q h, f h P, f h h Secant line f h f From the graph, we see that the slope of secant line PQ is m PQ change in y Rise f h f. change in Run h Difference Quotient f h f m h Eample 4: Let f 5, find the difference quotient and simplify your answer Solution: For clarity, we will solve this eample step by step. f ( ) 5 5 5 f h h h 5 h h h 5 f h f h h h f h f h h h 5 f h f h h h h h ( ) f h f h( h ) m h h h 5 Note 1: The simplified form of the numerator f h f should have h as a common factor. Note : Review your work if all of the terms of the original functions f() don t cancel in the final epression of the difference quotient. -79-
Let f and g 4 5 Section.5 Algebra of Functions and their Composites Eercises.5. Find the following. f 1... 4. 5. 1 g f g f g f g f g f 6. f g 7. f g 8. g f 0 9. 10. g g f 1 Let f 4 and g. Find the following. f 11. 1. 1. 14. 15. g f g f g f g f g f g 16. 1 17. 18. 0 19. 1 0. 4 g f f g f g g f Let f 4 1 and g. Find the following. 1. f g. g f. f f 4. g g f g g f f f 8. g g0 5. 1 6. 7. 1 Let f and g 1. Find the following. 9. f g 0. g f 1. f f. g g f g g f f f 6. g g1. 4. 0 5. In eercises 7 44, find the difference quotient of each given function and simplify the answer completely. 7. f 5 8. f 4 9. f 5 4 41. f 1 4. f 1 4. f -80-40. f 7 44. f 4 1
Chapter Analysis of Graphs of Functions Answers - Eercises.1 1. Odd. Even. Even 4. Neither 5. Odd 6. Neither 7. Even 8. Odd 9. Neither 10. Even 11. Odd 1. Odd 1. y-ais 14. Origin 15. Origin 16. Origin 17. y-ais 18. All 19. All 0. All 1. All. All. -ais 4. y-ais Answers - Eercises. 1. g. g. g 4. g 4 5. g 1 6. g 4 1 7. g 5 4 8. g 1 9. 10. 11. 1. 1. 14. 15. 16. 17. 18. 19. 0. 1. g 6. g -81-1 5. g 1 4 4. g 4 4 1
Answers to Chapter Problems Answers - Eercises. 1. 1,9. 1,5. 4,1 6., 1. 1. 7. 6, 5 5 8, 17. 1,7 18. 9,7 19. 1,4 4. 14. 8, 5. 1 1, 8. 9. 10. 11. 7, 8 1 0.,1 15. 1 16. 1. 6,. 10 8,4. 4. 5. 6. 7. 5 8. 9., 7, 0., 71, 1., 1, 9.,, 4. 5, 5, 7 5., 1, 5.,5 5, 6.,, 7. 8. 9. 40. 41. a),1, b), 1,, c),1 4. a) 4,, b), 4,, c) 4, 4. a) 5.5,.5, b), 5.5.5,, c) 5.5,.5 45. a) 6.5, 4.5, b), 6.5 4.5,, c) 6.5, 4.5 44. a), b), c) 46. a), b), c) Answers - Eercises.4 1. 1, -8,. -15, -7, 1. 1,, 5, 8 4., -4, 1, 5., 1, 7 6. 0, -5, 1 7. 4, 1, 4 8. 0, -7, -9-8-
Chapter Analysis of Graphs of Functions 9. 0, -,, -5, 11 10. 0, -6, 7, -8, 5 11. -5, -7,, 1, 1. -, -4, 1,, 6 1. -1, -5, 1,, 8 14. -, -, 1, 8, 10 15.,, 1, 7, 8 16. -9, -5, -4, 1, -1 17. -, 1,, 4, 9 18. -9, -5, -, -1, 1) 6 ) 8 ) 8 15 Answers - Eercises.5 4), 5 4 5 4 5) 8 6) 1 7) 4 8) 8 9) 17 10) 1 11) 1 1) 7 1) 5 14) 4 ; 1 15) 7 16) 8 17) 4 18) 7 19) 0) 1 1 1) 8 11 ) 8 1 ) 16 5 4) 4 9 5) 19 6) 15 7) 1 8) 9 9) 9 6 0) 10 1) 6 6 ) 9 4 ) 4) 10 5) 6) 5 7) 4 8) 9) 5 40) 41) h 4) h 4) h 44) 4 h -8-