9 x and g(x) = 4. x. Find (x) 3.6. I. Combining Functions. A. From Equations. Example: Let f(x) = and its domain. Example: Let f(x) = and g(x) = x x 4

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1 1 3.6 I. Combiig Fuctios A. From Equatios Example: Let f(x) = 9 x ad g(x) = 4 f x. Fid (x) g ad its domai. 4 Example: Let f(x) = ad g(x) = x x 4. Fid (f-g)(x) B. From Graphs: Graphical Additio. Example: Graph (f+g)(x) usig the graph below.

2 II. Compositio of Fuctios ( f g)( x) = f(g(x)) Example: Let f(x) = 3x-5 ad g(x) = a) Fid f(g(x)). b) Evaluate ( f g)( ). 3 x. c) State the domai of ( f g)( x). 1 Example: Let f(x) =, g(x) = 3 x, ad h(x) = x. x Fid ( f g h)( x). Example. Express H ( x) 1 x i the form ( f g)( x).

3 3 Example: A stoe is dropped i a lake, creatig a circular ripple that travels outward at 60 cetimeters per secod. a) Fid a fuctio g that models the radius as a fuctio of time. b) Fid a fuctio f that models the area of the circle as a fuctio of time. c) Fid f g. What does this fuctio represet?

4 4 Example: A prit shop makes bumper stickers for electio campaigs. If x stickers are ordered (x < 10,000), the the price per bumper sticker is $( x ), ad the total cost of producig the order is $( 0.095x x ). a) Reveue = (Price per item) X (Number of items sold) Express R(x), the reveue from a order of x stickers, as a product of two fuctios of x. b) Profit = Reveue Cost. Express the profit P(x) o a order of x stickers as a differece of two fuctios of x.

5 5 3.7 I. Combiig Fuctios Fuctio f Iverse fuctio f -1 Domai Domai Rage Rage f(x) = y f 1 ( y) x poit (a,b) poit (b, a) A fuctio must be oe-to-oe to be ivertable. Oe-to-oe: If f ( x1 ) f ( x), the x 1 x. Pass HLT Every icreasig fuctio is 1-1 Every decreasig fuctio is 1-1 Example: Sketch f ( x) x x 6 usig your calculator to determie whether f(x) is 1-1.

6 6 Example: G(x) = (x 1) is graphed below. a) Restrict the domai so that the fuctio is 1-1. b) Graph the iverse fuctio. Example: Sketch f(x) = x 1. Use this to sketch f 1 ( x ). 3 Example: Determie whether h(x) = x 8 is 1-1.

7 7 Steps to fid a iverse fuctio. 1. Replace f(x) by y.. Switch x ad y. 3. Solve for y. 4. Replace y by f 1 ( x ) Example: Fid the iverse of f ( x) 4 x, x 0.

8 8 To verify that two fuctios are iverses, show f(g(x)) = g(f(x)) = x, for all x. Example: Verify that x 5 f ( x) ad 3x 4 5 4x g( x) are iverses. 1 3x Example: If f(5) = 18, f 1 (18) Example: If f 1 (4), f() =

9 9 4.1 Name Equatio Vertex at Quadratic Form f(x) = ax bx c b, f a b a Stadard form f ( x) a( x h) k (h,k) Example: Fid the vertex of s 1.s 16. To Graph 1. Express the equatio i stadard form.. Fid the vertex. 3. Fid the x-itercept(s). 4. Fid the y-itercept. 5. If a > 0, the graph is CCU If a < 0, the graph is CCD 6. Coect the poits.

10 10 Example: Graph f ( x) x 4x 4. State the domai ad rage.

11 11 Example: Fid a fuctio whose graph is a parabola with a vertex at (3, 4) ad passes through the poit (1, -8). Example: Fid the maximum or miimum of f ( x) 1 x x. a) O the graphig calculator. b) By had. c) Compare the results.

12 1 Example: A ball is throw across a playig field from a height of 5 feet above the groud at a 0 45 agle to the horizotal at a speed of 0 feet per secod. The ball s path is modeled by: y 3 x 0 x 5, x = horizotal distace traveled. a) Fid the maximum height of the ball. b) Fid how log it takes the ball to hit the groud.

13 13 Example: A corral has,400 feet of fecig to fece i a rectagular horse coral. a) Fid a fuctio that models the area i terms of width. b) Fid the dimesio of the rectagle that maximizes the area of the corral.

14 14 4. ad 4.3 Form I: Polyomial of degree P( x) a x... a 0 A. Write i descedig powers Polyomials are cotiuous (o breaks or holes) Smooth curve (o cusps) Number of max/mis = degree 1 Degree = umber max/mis + 1 At most At least Ex: Graph each polyomial ad the determie the umber of maximums ad miimums a) Y 1.x 3.75x 7x 15x 18x b) Y (x ) 3 B. Ed tail behavior. Eve Degree Odd Degree a 0 a 0

15 15 C. To graph a polyomial of degree : 1. Graph x. Apply trasformatios 3. Label the ed tail behavior. Example: Graph P ( x) x 4 Type II: Factored Form 1 P ( x) a( x z ) ( x z )...( x z k 1 k ) To graph: 1. Degree = Sum(expoets) ; apply ed tail behavior. Fid the zeros of the polyomial 3. If r is a zero of eve multiplicity, the graph touches the x-axis at root, r If r is a zero of odd multiplicity, the graph crosses the x-axis at root, r. 4. Fid the y-itercept 5. Graph Example: Graph each polyomial. 3 a) P ( x) ( x 1) ( x ). b) P ( x) (x 1)( x 1)( x 3) 4 3 c) P ( x) x 3x x

16 16 Itermediate Value Theorem. If P is a polyomial ad P(a) ad P(b) have opposite sigs, the there exits at least oe value c betwee a ad b such that P(c) = 0. Example: A ope box is to be costructed from a 0cm by 40cm piece of cardboard by cuttig squares of equal legth x from each corer. a) Express the volume (V) of the box as a fuctio of x. b) What is the domai, kowig that legth ad volume are positive? c) Draw a graph of V(x). Estimate the maximum value.

17 17 II. Log Divisio. Divisio Algorithm Divided = Divisor * Quotiet + Remaider P(x) = D(x)*Q(x) + R(x) 4 3 Example: P ( x) x x 4x D ( x) x 3

18 18 III. Sythetic Divisio Use whe divisor is of the form D(x) = x c or D(x) = x (-c) = x + c. Steps: 1. c coefficiets of P(x) writte i descedig powers -c coefficiets of P(x) writte i descedig powers 3 Ex) P ( x) x 4x 6x 1 D ( x) x 1. Write i the coefficiets. Use zero as a place holder for missig terms. 3. Carry dow the first coefficiet below the divisio symbol. 4. Multiply the divisor by the first coefficiet. 5. Write the product above the lie ad i the first colum to the right. 6. Add the ext coefficiet to the product. Record the sum below the lie. 7. Repeat the multiply ad add steps util the last row is complete. A. Factor Theorem: P(x) = (x c) Q(x) + R(x), If R(x) = 0, the the polyomial is factored ad x = c is a zero of the polyomial. Example: x 5 x x

19 19 B. Remaider Theorem Evaluate P(x) at x = c meas Example: Evaluate P(c) at c = -3 whe P ( x) x 7x 40x 7x 10x 11 C. Factor Theorem (Revisited) x c is a factor of P(x) if ad oly if P(c) = 0. Steps: 1. Check to see if P(c) = 0. If so, the R(x) = 0 i sythetic divisio ad x c is a factor of P(x). Why?. Perform sythetic divisio. 3. P(x) = (x c)q(x) Factor Q(x)

20 0 Example: Show 1 3 x is a factor of P ( x) x 7x 6x 5. Example Show c = - ad c are zeros of 3x x 1x 11x 6. The fid all other zeros. 3

21 1 To fid a polyomial with specific zeros: Let z 1 = root 1, z = root,, z = root. The 1 P ( x) a( x z1 ) ( x z)...( x zk ) k Example: Write a polyomial of degree 5 with zeros at -, -1, 0, 1, ad.

22 4.4 Goal: Factor P( x) a x... a0 Use the factored form to fid solutios to P(x) = P( x) a x... a0 = 0 This will result i the zeros/x-itercepts/factors of the polyomial. Cocept: The complex zeros ca be: 1. Real ad Ratioal 0 = 4x 9. Real ad Irratioal 0 = x 3. Imagiary 0 = x 4 Steps to fid the zeros: 1. Total umber of complex zeros = Degree of the polyomial.. Descartes Theorem Example: Fid the umber of real zeros for each polyomial below. 4 3 a) P ( x) x x x x 1 3 b) P ( x) 4x 8x 11x 15

23 3 3. Ratioal Zeros Theorem If P( x) a x... a0 has degree 1ad iteger coefficiets p factors of a 0 p The for x, x is a potetial zero of P(x). q factors of a q 3 Example: List the possible zeros of P ( x) 4x 8x 11x Choose which of the potetial zeros are ideed zeros. Example: Choose the zeros for P ( x) 3 4x 8x 11x 15. Method I: Graph. Method II: Sythetic Divisio Method III: Use the table o your calculator.

24 4 5. Use sythetic divisio to write the polyomial i fully factored form. 3 Example: Factor P ( x) 4x 8x 11x 15.

25 5 Useful Commets: 1. P(x) ca ot have more real zeros tha its degree.. The total umber of complex zeros = degree of the polyomial. 3. A polyomial fuctio with real coefficiets ad odd degree has at least oe zero. 4. Upper Boud/Lower Boud Theorem Divide P(x) by p q usig sythetic divisio. Example: Show that a = -3 ad b = 5 are upper ad lower bouds for P( x) 4 x 3 x 9x x 8.

26 6 4.5 Complex Numbers z = a + bi If b = 0 : If a = 0: z = 3 + 4i Complex cojugate of z = a + bi is z =a bi. Ex: z = 3 + 4i For 1 k P ( x) a( x z1 ) ( x z)...( x zk ) z i are the complex zeros, i = 1 k (degree of polyomial) (x - z i ) are the factors I. Fudametal Theorem of Algebra *If a polyomial P( x) a x... a0 has positive degree ( 1), a 0, ad complex coefficiets (a + bi) * The P(x) has at least oe complex zero. 4 Ex: x 4 II. This theorem allows every polyomial of positive degree to be expressed as a product of polyomials of degree oe. Complete Factorizatio Theorem For all * P( x) a x... a0 *, there exist complex umbers 1 k a, z 1, z k such that P ( x) a( x z1 ) ( x z)...( x zk ). Commet: k = = degree of the polyomial.

27 7 III. Zero s Theorem Every polyomial * P( x) a x... a0 * has exactly complex zeros. Zeros ca be: o Pure Real o Pure Imagiary o A combiatio of real ad imagiary Zeros that occur with multiplicity m are couted m times. If m = odd, the graph flattes ad crosses the x-axis. If m = eve, the graph bouces off the x-axis. Cojugate Zeros Theorem If z = a + bi is a complex zero of P(x), the z =a bi is also a complex zero. Ex: P(x) is of degree three ad has oe imagiary zero. What do you kow about the other zeros of this polyomial? Example: Fid all solutios to ix x i 0. Example: Fid a polyomial with iteger coefficiets of degree three with zeros at -3 ad 1 + i. Example: Factor each zero. 5 3 P( x) x 6x 9x ad the fid all complex zeros. State the multiplicity of

28 8 6 Example: Factor P ( x) x 64 a) ito liear ad irreducible quadratic factors with real coefficiets. b) Ito liear factors with complex coefficiets. 4 3 Example: Fid all zeros of P ( x) 4x x x 3x 1.

29 9 I. Notatio ad defiitios Vertical Asymptotes x c meas x approaches c from the left. Ex) x c meas x approaches c from the right. Ex) Defiitio: If, as x approaches some umber c, the values y approach ifiity, the the lie x = c is a vertical asymptote. (Rus parallel to the y-axis).. Horizotal Asymptotes x meas x icreases without boud. x meas x decreases without boud. Defitio: If, as x approaches positive or egative ifiity, the values y approach a fixed umber L, the the lie y = L is a horizotal asymptote. (Rus parallel to the x-axis).

30 30 3. Oblique Asymptotes If the asymptote is ot parallel to the x-axis or the y-axis, the the asymptote is called oblique. x, y mx b x, y mx b For this graph: x 0, y x 0, y Notice that if m = 0, the f(x) = mx+b becomes f(x) = b. So the slat asymptote becomes a horizotal asymptote.

31 31 II. Ratioal Fuctios r ( x) P( x) Q( x) ax b x d d a b 0 0 mx b R( x) Q( x) ( x ( x z1)( x z )( x 1 z)...( x z )...( x z) z ) d A. Vertical Asymptote: Zeros of the deomiator. If zi is a zero of the deomiator of r(x) i lowest terms, the x = z i is a vertical asymptote. Example: 5 x has a vertical asymptote at x =. B. Horizotal Asymptote: For r( x) P( x) Q( x) a x b x d d a b Proper Form ( < d) y = 0 is the horizotal asymptote (the x-axis). Example: 5 x has a horiztoal asymptote at y = 0.. Improper Form. Divide P( x) R( x) r ( x) mx b. Q( x) Q( x) Case I: If = d, the f(x) = 0x+b = b, ad y a b d is the horizotal asymptote. Example: x x 4 ( x). x x S CaseII: If > d, the f(x) = mx+b ad mx+b is a oblique asymptote. Example: 3 x x 3x 4

32 3 Commet: A graph ca cross a horizotal asymptote. (Does ot happe ear ifiity). C. To graph 1. Factor. Fid ay horizotal or oblique asymptotes (compare degrees) 3. Fid the x-itercept (zeros of the umerator). 4. Fid the vertical asymptotes (zeros of the deomiator) 5. Graph WARNING: If there is a commo factor i the umerator ad deomiator, the this poit forms a hole o the graph. Ex: r ( x) (x (x 5)(4x 5)( x 3) 3)

33 33 Some Geeral Shapes of Ratioal Fuctios

34 34 Example: Graph the equatio below. a) 3 x r ( x) x x 1