2.4 Operations with Functions

Transcription

1 4 Operations with Functions Addition and Subtraction of Functions: If we are given two functions f( and, they may be combined through addition and subtractionas follows: ( f ± g)( f ( ± Example: Add the functions f ( + and g ( 5x 9 Solution: Using the above definition ( f + g)( f ( + f ( + (3x + ) + (5x 9) + + 5x 9 8x 7 Example: Given f ( x + 3 and g ( x + x 5 find ( f + g)(4) Solution: Using the above definition ( f + g)(4) f (4) + 4) f (4) + 4) ((4) + 3) + ( (11) + (7) 18 + (4) 5) Example: Find the difference ( f g)( given f ( 4x + and g ( x + 3 Solution: Using the above definition ( f g)( f ( f ( (3x 4x + ) ( x + 3) 4x + + x x 1 Example: Find the difference ( f g)( ) given f ( 4x + 1 and g ( x 1 Solution: Using the above definition ( f g)( ) f ( ) ) f ( ) ) (4( ) + 1) (( ) (4) (7) 3 1)

2 Multiplication of Functions: If we are given two functions f( and, they may be combined through multiplication as follows: ( f g)( f ( Example: Multiply the functions f ( + 5 and g ( 4x 7 Solution: Using the above definition ( f g)( f ( f ( (3x + 5)(4x 7) 1x 1x 1x + 0x 5 x 5 To multiply the above functions the FOIL method of multiplication was used Example: Find the value of ( f g)(3) f ( x + 1 and g ( x 5x + Solution: Using the above definition ( f g)(3) f (3) 3) f (3) 3) ((3) + 1)(3 (7)( 4) 8 5(3) + ) Division of Functions: If we are given two functions f( and, they may be combined through division as follows: f g ( f ( f Example: Given f ( + 5 and x + find ( g Solution: Using the above definition f g ( f ( f ( 3x + 5 x +

3 Composition of Functions: In many applications it is necessary to use the output from one function as the input of another function as demonstrated in the following example Example: Given the functions f ( x + 5 and g ( 8 do the following: 1 Find f () Solution: Substitute x into the function f ( x + 5 f ( x + 5 f () + 5 f () 7 Evaluate the function 8 using the output value of f ( ) 7 Solution: The output value of f ( ) 7 is 7 8 7) 3(7) 8 7) 1 8 7) 13 Therefore, by starting with an input of, we eventually ended up with an output of 13 Here is a diagram (mapping) that demonstrates this relationship A composite function is one function that will take the place of two or more other functions In the above example we started with an input of and after using two functions ended up with an output of 13 A composite function would do the same using only one function Example: Show that the function h ( + 7 could be a composite function of the two functions f ( x + 5 and 8 Solution: In the above example, the original input was, so if we use this as the input we need to end up with an output of 13 h( + 7 h() 3() + 7 h() 13

4 Here is a mapping of this relationship Now, in order for h( + 7 to be a composite function of the other two functions, this relationship needs to hold true for any input value In the diagram below it can be seen that it holds true for at least 3 additional input values 3, 5, and 10 Our goal is to eventually show that this relationship holds for all x- values (input values) Evaluating a Function at a Function: Example: Evaluate the function f ( 4x 7 at g ( x + 1 Solution: Evaluating the function f ( at g ( x + 1 means we are replacing the variable x with the entire function Therefore, f ( 4x 7 ) 4(x + 1)) 7 ) 8x ) 8x If we let h ( ) then h( 8x is the composite function of functions f ( 4x 7 and g ( x + 1 The following mapping demonstrates this fact Take note that the function is used before f( The reason for this will be discussed shortly

5 Finding Composite Functions: Based on the work we did in the previous example, we can find the composite of two functions by simply evaluating one function by replacing its variable with the rule of another function Example: Given the functions f ( x + 3 and 5x + find ) Solution: The notation ) suggests that we are substituting into the function f( to obtain ) f (5x + ) f ( x + 3 ) (5x + ) + 3 ) 10x ) 10x + 7 Example: Given the functions f ( + 1and x 1find ) Solution: The notation ) suggests that we are substituting in place of the variable in the function f( to obtain ) f (x 1) f ( + 1 ) 3(x 1) ) 3(4x ) 1x + 1 4x + 1) + 1 1x + 4 Example: Given the functions f ( 4 x and x find g ( f ( ) Solution: The notation f ( ) suggests that we are substituting f( in place of the variable in the function to obtain g ( f ( ) 4 x f ( ) (4 f ( ) 16x Example: Given the functions f ( 5x + and 3 x find f (4)) Solution: The notation f (4)) suggests that we are substituting 4) in place of the variable in the function f( It will be helpful to first find 4) 3 x 4) 3 4 4) 5 Now substitute g ( 4) 5 in place of the variable in f( f ( 5x + 4)) 17 4)) 5( 5) +

6 Composite Function Notation: The notation for a composite function can be written in two ways: 1 ) which reads as f of g of x and ( f οg)( which reads as f follows g of x Therefore, ) ( f οg)( These two forms of a composite function are completely interchangeable Example: Given the functions f ( x and 4x + 1find ( f ο g)( Solution: The notation ( f οg)( ) and suggests that we are substituting into the function f( to obtain ( f ο g)( f (4x + 1) f ( x ) (4x + 1) ) (16 x ) + 8x + 1) + 16x 1 Order of Composition: It is important to understand that a composition of functions is not commutative, that is the order of composition is important This means that: ) f ( ) Example: Given the functions f ( x + 5 and + find the compositions ) and f ( ) and compare the results Solution: ) f (3x + ) ) (3x + ) + 5 ) 6x + 9 f ( ) x + 5) f ( ) 3(x + 5) + f ( ) 6x + 17 It can be seen that ) f ( )

7 4-Applications Example: The cost in dollars that a company spends for installing x alarm systems per month are given by C ( 600x The company needs to earn a profit given by the function P ( 0x + 400x 4000 Determine the revenue function that is required in order to satisfy this condition Find the revenue for installing 45 alarm systems per month? Solution: We will use the profit equation P( C( P( C( P( + C( ( 0x 0x + 400x 4000) + (600x ) x To find the revenue when installing 45 alarm systems, find 45) 0x 45) 0(45) x (45) 45) 40, ,000 45) 94,500 The company will earn revenue of $$94,500 when it installs 45 alarm systems Example: A factory has been selling 1000 color laser printers per year for $1500 each The company s market research indicates that for each $100 that the price is raised, sales will fall by 30 units Create a function that will determine the yearly revenue generated for every $100 increase in the price of a printer Solution: Revenue is determined by multiplying the selling price of an item by the number of items sold Let x the number of $100 increases We can then create a price function and a number sold function in terms of x The revenue function is therefore P( S( P( x S( x ( ( ,500,000 45,000x + 100,000x,000x 3,000x + 55,000x + 1,500,000

8 Example: Steve is a long haul trucker When he drives 300 miles in a day he takes 6 hours, however for every 100 additional miles Steve drives he finds that the time it takes him increases by 3 hours Create functions that represent Steve s driving distance and driving time in terms of 100 mile increments Then create a function that represents Steve s driving rate What is Steve s rate if he drives 700 miles in a day? Are there any restrictions on how far Steve can drive in a day? Solution: Let x equal the number of 100 mile increases Steve drives per day Then we can create the following functions: D( x T ( 6 + 3x Since D RT we can rearrange the formula to obtain the rate function D( T( x 6 + 3x To determine Steve s rate at 700 miles, we let x 400 and substitute this value into the function x 6 + 3x (4) 6 + 3(4) Therefore if Steve drives 700 miles in a day his driving rate will be approximately 388 miles per hour Example: If hamburgers are $10 each, then C( 1 0x gives the pre-tax cost in dollars for x hamburgers If sales tax is 5%, the T ( c) 1 05c gives the total cost when the pretax cost C Write the total cost as a function of the number of hamburgers How much will 3 hamburgers cost including the tax? Solution: The formula may be determined by creating a composite function ( T ο C)( To find the cost of 3 hamburgers, find ( T ο C)(3) The cost of three hamburgers, including tax is $378 ( T ο C)( ) T ( C( ) T (10 105(10 16x ( T ο C)(3) 16(3) 378

9 Example: The sail area displacement ratio S measures the sail power available to drive a sailboat: Where A is the sail area in square feet and S ( y) A y y( d) d 64 3 Where d is the displacement in pounds Assuming that the sail area is 6500 square feet, write S as a function of d and simplify it Are there any restrictions on the displacement? If so, what is the restriction? Solution: First, create the composite function S ( y( d )) A A 6500 S( y) 3 y d ( d ) , ,000 3 ( d ) ( d ) d The restrictions on the domains of the original functions are: D : S( y) y 0 and D : y( d ) d R 3 The domain of the composite function must be at least as restrictive as either of the original function, therefore, the domain of the composite function is: D : S( y( d)) d >

10 Example: Southern Sod will deliver and install 0 pallets of St Augustine sod for $00 or 30 pallets for $300, not including tax 1 Write the cost function as a linear function of x, where x is the number of pallets Write a function that gives the tax on x pallets if the tax rate is 9% 3 Write a function for the total cost as a function of x, where x is the number of pallets 4 Use the total cost function to determine the total cost of purchasing 10 pallets of sod Solution: Write the cost function as a linear function of x, where x is the number of pallets Create the ordered pairs (0, 00), and (30, 300) m y y 1 m( x x1) y ( x 0) y 100x + 00 C( 100x + 00 Write a function that gives the tax on x pallets if the tax rate is 9% T ( 009( C( ) T ( 009(100 x + 00) T ( 9x + 18 Write a function for the total cost as a function of x, where x is the number of pallets TC( C( +T() is the sum of the cost of purchasing x pallets plus the tax on x pallets TC( C( + T( TC( (100 x + 00) + (9x + 18) TC( 109x + 18 Use the total cost function to determine the total cost of purchasing 10 pallets of sod TC( 109x + 18 TC(10) 109(10) + 18 TC(10) 1308 It will cost $1,30800 to purchase 10 pallets of sod