The algebra of functions Section 2.2

Transcription

1 Te algebra of functions Section 2.2 f(a+) f(a) Q. Suppose f (x) = 2x 2 x + 1. Find and simplify. Soln: f (a + ) f (a) = [ 2(a + ) 2 ( a + ) + 1] [ 2a 2 a + 1 ] = [ 2(a a) a + 1] 2a 2 + a 1 = 2a a a + 1 2a 2 + a 1 = a = (2 + 4a 1 ) So f(a+) f(a) = = 2 + 4a 1 (2+4a 1) f(a+) f(a) 1 Q. Suppose f (x) = x. Find and simplify. Soln: f (a + ) f (a) = 1 (a+) 1 a a (a+) = (a+)a = a(a+) So f(a+) f(a) = a(a+) = 1 a(a+) f(a+) f(a) Q. Suppose f (x) = x. Find and simplify. Soln: f (a + ) f (a) = a + a f(a+) f(a) a+ a = ( a+ a )( a+ + a ) = = (a+) a 1 = = ( a+ + a ) ( a+ + a ) ( a+ + a ) ( a+ + a ) So

2 Q. Te birtrate of an endangered species of wales in year t is f (t) wales/year. Tis species of wales is dying at te rate of g(t) wales/year in year t. Wat does te function F (t) = f(t) g (t) represent? Ans. F (t) = f(t) g (t) represents te rate of increase of te wale population in year t. Q. Te number of BPL sares tat Sam owns is given by f (t). Te price per sare of BPL at time t is g (t) dollars. Wat does te function (t) = f (t)g(t) represent? Ans. (t) = at time t. te net dollar value of te BPL sares owned by Sam Q. Te total cost incurred by time t in te production plastic cups is f (t) dollars. Te number of plastic cups produced by time t is g (t) units. Wat does te function (t) = f (t)/g(t) represent. Ans. It represents te average cost per cup produced by time t. Q. Te number of cars running in Lawrence at time t is given by f (t). Carbon monoxide pollution coming from tese cars is given by g(x) parts per million, were x is te number of cars being operated in Lawrence. Wat does te function g f represent. Ans. ( g f )(t) represents te total carbon monoxide pollution coming from te cars running in Lawrence at time t.

3 Q. A division of Capman Corporation manufactures a pager. Te weekly fixed cost for te division is $ 20, 000, and te variable cost for producing x pagers/week is V (x) = x x x dollars. Te company realizes a revenue of R (x) = 0.02x x (0 x 7500) dollars from te sale of x pagers/week. a. Find te total cost function. b. Find te total profit function. c. Wat is te profit for te company if 2000 units are produced and sold eac week? Soln. a. Te cost function for producing C (x) = x x x x pagers per week is dollars b. Profit = Revenue - Cost. So te total profit function from te sale of x pagers per week is P (x) = R(x) C (x) = [ 0.02x x] [ x x x] = x x x dollars/week c. Wen te production is is P (2000) =? 2000 units per week. Te weekly profit

4 Functions and Matematical Models. Section 2.3 A polynomial function of degree n is a function of te form f (x) = anx n + a x n 1 n a x a 1 x + a 0 Te numbers a n, a n 1,..., a 1, a 0 are called te coefficients of te polynomial function. a n is called te leading coefficient. a 0 is called te constant term. Example : f (x) = 2x 5 3x x 3 2x 6. Degree = 5, constant term = 6, leading coefficient = 2 Example : f (x) = x Degree = 1000, constant term = Example : f (x) = 1 Degree = 0 since = 1. f (x) = ( 1)x0,Constant term = 1, leading coefficient Domain of a polynomial function is (, ). A polynomial function of degree 1, is called a linear function. It is of te form a 1 x + a 0 or m x + b. 1 Examples: f (x) = x + 5, f (x) = 2 x + 2, y = m x + b

5 Q. Does te relation 3y+2x+5=0, define y as a linear function of x.? Soln: 3y+2x+5=0 3 y = 2x y = 3 x So te given relation does define a linear function. Te grap of te function f (x) = m x + b, is te straigt line y = m x + b in te xy-plane. Tis line as slope m, and y-intercept b. Hence for a function f (x) = m x + b, m is called te slope, and b is called te y-intercept. Equation of a line: To find te equation of a line y=mx+b, we need to determine te slope m and te y-intercept b. Q. Find te equation of a line passing troug (0,1) and (3,4). Soln. Te slope is. m = = 1 Let te equation be y=x+b. Ten (0,1) must satisfy tis equation.

6 So 1=0+b. So b=1. Hence te equation of te line is y=x+1. Q. Find te equation of a line wit slope m=2, and wic passes troug (3,4). Soln. Let te equation of te line is y=2x+b. Ten (3,4) must satisfy tis equation. So 4=2(3)+b. So b=-2. So te equation of te line is y=2x-2. Q. Find te equation of line wit slope m=2, and y-intercept 3. Soln. y=2x+3 Remark: Given someting like 3x+4y=1, ten y=-3/4x+¼. Hence y is a linear function of x and represents te line in te xy-plane wit slope -¾ and y-intercept ¼. Te form y=-3/4x+¼ is called te standard form of te equation of te line. Q. How do you determine if two equations like 3x+4y=1, and -9x-12y=-3 represent te same line in te xy-plane. Soln. Two equation represent te same line if and only if tey ave te same standard form. Tat is tey ave te same slope and te te same y-intercept. For example te above two equation bot ave te standard form y=-3/4x+¼, ence tey represent te same line. Q. Do te relations functions. 2x 2 8 y + 4 = 0, 3 x + 4 y = 0 define linear 2 Soln. Te first relation gives y = x , wile te second relation gives y = 3 4 x. Te first function is a quadratic

7 function, wile te second function is a power function. It is not a linear function in bot cases. Quadratic functions: A polynomial function of degree 2 is called a quadratic function. It as te form y = ax 2 + b x + c. Examples: y = 2x x + 1, y = x 2 + x + 1

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9 Rational and Power functions A rational function is simply a ratio of two polynomials. Examples:, x 1 x 2 +1 x 2 1 Te domain of a rational function consists of all real numbers except tose wic are zeroes of te denominator. Domain of x 2+1 x2 1 is { x = / ± 1 } Any function of te form can be any real number. f (x) = x r is a power function, were r Examples: x, x 2/3, x 1 2 Q Wat is r in te above examples. Soln. ½,-⅔,-2 respectively. Many of te functions we will encounter will be combinations of 1+x2 tese simple functions. For example 1 x 2, 2x x 2 +1

10 Identify te type of te given functions: Polynomial? Rational? Power? If it is a polynomial function find te degree. Find te domain in eac case. f (x) = 0, 4x x 1 x ,, 2x x, x 0.5 x 3 Q. Te oxygen consumption in mil/lb/min for a person walking at x 5 mp is approximated by te function f (x) = x x + 10 (0 x 9 )

11 Te oxygen consumption for a runner at x mp is approximated by te function g (x) = 11x + 10 (4 x 9 ) (a) Sketc te graps of f and g. (b) At wat speed is te oxygen consumption te same for a walker as it is for a runner. (c) Wat is te level of oxygen consumption at tat speed. Wic of tese two graps is te answer to (a).