Reviewer in General Mathematics, Study notes of Mathematics

Download Reviewer in General Mathematics and more Study notes Mathematics in PDF only on Docsity! Reviewer in General Mathematics ~ Representation of Function (multiple choice) 15 pts. * Relation- a set of ordered pairs, the domain of a relation is the set of first coordinate. Relation {(1,c) , (5,a) , 8,b) } Domain www.tallwarelouse. cor * Function- is a relationship in which each element of the domain corresponds to exactly one element of the range. Relation #1 Relation #2 is a function is not a function <(1, 6), (a), (8, b)} (1,6), (a), (5, ) } ance He sae was (5) hes 2 ifjorent ¥ valet *Characteristics of Function* 1. Each element in the domain X must be matched with exactly one element in range Y. 2. Some elements in Y may not be matched in any element in X. 1. Mapping Diagram 2. Table of Values 3. Graph- represents a function if and only if no vertical line intersects the graph in more than one point. Vertical Line Test - Agraph represents a function if and only if no vertical line intersects the graph in more than one point. 4. Rule of Correspondence fix—> x7H1, x=1,2,3,4 5. Equation The rule of correspondence can be described by the equation ext also be written f(x)=x7+1 Function Not Function Note: When finding the domain and range of a function involving: 1. Aradical with an even index: Radicand must be non-negative. Hence the radicand must be greater than or equal to zero. 2. A fraction: Denominator must not be equal to zero. «Linear Function (solving) 10 pts. - a function is a linear function if f(x)=mx+b, where m and b are real numbers and f(x) are not both equal to zero. y=mx+b —» slope-intercept form Ax+By=C — standard form y-Yi=m(x-x,) —point-slope form Yo7Vi1—» =———_ Slope XQ Xy *Problem* To sell more T-shirts, the class needs to charge a lower price as indicated in the following table: 500 P 540 900 P 460 1300 P 380 1700 P 300 2100 P 220 2500 P 140 The price for which you can sell the x printed T- shirts is called the price function p(x). _Yo7¥1 460-540 -qU J] ox, ™ 900-500 ™~4op M5 Slope y-y:=m(x-x;) -1 y-540=—— (x-500) -1 y- 540=—"x +100 -1 y= x #1004540 yooty +640 is *Practice Exercise* 1. What is the slope passing through the points (4,-6) and (8,-5)? 2. What is the slope and y-intercept of y=4x-6? 4.Graph the equation y=3x-2 using the slope- intercept form. wo . Graph the equation 2x-3y=6 using the x and y Solution: Graph: 6. Write the equation of the line passing through the point (2,5) with the slope of 3. 7. Write the equation of the passing through the points (-3,1) and (2,-4). Solution: Graph: 8. Write the equation of the line passing through the points (3,-2) and parallel to the line 2x+5y-3=0. 3. Graph the equation x=2 and y=3? Solution: Graph: applies to a certain interval of the main function's domain. Example: (x)= 2x-2, If x<1—> linear f(x)= x?-2x+1, If x21 —>quadratic - the compound function given is defined by two equation. One equation gives the value of f(x) when x is less than or equal to 1, and the other equation gives the value of f(x) when x is greater than 1. aye le *% One-to-One Function - a one-to-one function is ae which for each value of y in the range of it, there is just one value of x in the domain of f such that y=f(x). Example: f g 1 1 —> 5 — ° 2——*+7 (a) (b) - Ina, there are two values in the domain that are both mapped onto 5 in the range. Hence, the function f is not one-to-one. However, in (b), for each output in the range of g, there is only one input in the domain that gets mapped onto it. Thus, g is one-to-one function. Example: 1. f(x)=3x-5 one-to-one e If X, and X, are real numbers such that f( X)=f(X2), then 3X1-5=3X55 4x,=B x, X\=X> 1 2. Function x, xx , etc. one-to-one x 11 5. 57. 7 e If X,#X, then x}#xz, X}#X3, 4 XX 3. ton +1 one-to-one e If X, and X, are real numbers such that f( X1)=f(X,), then Tx, 3 ated xs 3 Not one-to-one one-to-one one-to-one Example: Determine if each function is one-to-one function. a. f(x)= 5x-7 5X 1-7= 5Xy-7 b. f(x)= x2+2 *1=Xone-to-one one-to-one d. La Not one-to-one Ine-to-one * Evaluation of Functions: y=f(x) Example: 1. y=3xt1 f(x)=3xt1. 1 4(4)=3(4)41 (2) 3(2)+1 £(3)= 3(3)41 3 anata yale ata A x5 e — If f(x)=x+8, evaluate each a. £(4) +8 c. f(-x) b. f(-2)=-2+8 d. f(x+3)= x+3+8. ¢ Even Function - the function is an even function if and only if f(-x)= f(x) ¢ Odd Function - the function is odd function if and only if f(-x)= -f(x) Example: a. f(x)= x° f(-x)= -x° —odd function b. g(x)= 3x4-2x? al-x)-3(—x)*-2(-x? = 3(-x)(-x)(-x)(-X)-2(-x)(-x) = 3x2? even function c. (x)= x24 F(-x)= (-x)(-x)(-9)-4 f(-x)=-x*-1 —® odd function d. g(x)= 3x7-x 441 = 3Ex)(-x)-(-9)(90(-x) (x) 1. g(x)= 3x7-x4+1 —Peven function e. 2x°447 F(x) = 2(-x) (-x) (-x)(-%)(-X)(-) +4 (-X)(-X) f(-x)= 2x°+4x? —even function ' ‘ol® x45 if x>—4 ox? —Lifxed find: a. f(-5) b. (3) c. f(-2) f(-5)=2x7-1 £(3)=6x+5 ——f(-2)=6x+5 =2(-5)*-4 = 6(3)45 = 6(-2)45 = 2(25)-1 = 1845 [242x4 Lif x>2 & foo Ax—2if x<2 find: a. f(-1) b. (5) c. f(-4) f(-1)= 4x-2 #(5)= x742x4+4. f(-4)=4(-4)-2 = A(-1)-2 =(5)+2(5)+4 16-2 =-4-2 10+4 Fe==6] d. f(8) f(8)= x2+2x+1. = (8)°+2(8)41 64+ 16+ 1 “Operation of Function: Perform the indicated Operations: a. (4x+3) + (3x-2)_b. (2x7-3x-2)-(4x7+5x+3) 2 = 7x44 2x 7 3x-2 —4x°-5x-3 =-2x°-8x—-5 c. (2x-3)(x+4) d. (x7-5x+2)(3x+1) = 2x74BX-3X12 = 3x°15¢7H6x+x7-Sxt2 = 2x745x-12 = 3x°-14x74x+2 e. (V34x)(V3-x) —f. (2x?-9x-35)+(2x+5) =| A= pox 2x+5 241 9 x-35 =3-x? 22x7-5x wy = 14x35, 0 ® Law of Exponents 0 Product Rule Ex: 12x702x' (x? y "x4 y?) O Power Rule = (x ("= x" Ex: (x72 Ee] O Power of a Product Rule= (xy{"= mon xy Bx: (xyi7Ex? y* * Special Products O Square of Binomials Q__Cube of Binomials =x?-2xyty (xg3= x 3ixe*(-VIF-3x¥V 30x)lyg3=3xy° 32,3 (yet=y? ( 0 Product of Sum & Difference (x+y) &y) 2 O Square of Trinomial (2x+3y+27 (2xg2= 4x? (3yg?= 9y* (277= 4 2(2x)(3y)= 18xy 2(2x)(2)= 8x 2(3y)(2)= 12y = Ax AD y"+4+18xyt 8xt12y xe 3(x22(y)= 3x°y 30)(yE8=3xy° ¢ Rules in Division of Monomials m = ym x - == x™™" when men m x nn yn-m x when m<n m x = 1 when m=n x Example: 1. f(x) g(x) 5x-10 x2 a. f(x)+g(x) b. f(x)-g(x) = (5x-10)4{x-2) =(5x-10)-(x-2) = 6(4x7 442 = 6x-12 = 4x8 (fog)(x)= 24x7+2 (Fe)(x)= 6(x-2)} (f-g)(x)= 4(x-2) g(f(x))= 2(6x*+2) c. flx)eg(x) d. f(x)=8(x) (gof)()= 12,244 = (5x-10)(x-2) (£ (je 2X10 \g! x-2 = 5x7-10x-10x+20 = Sie) x#2 t = 5x°=20x+20 [t loa = 5, x#2 ToT 5 (x)*g (x)= x7-4x44 2. (x) g(x) x41 x7-5x+4 a.f(x)+g(x) - add b. f(x)-g(x) - subtract (x4) (25x44) Loeb 2aS tA) (Feg)(x)= x7-4x+3 (F+g)(x)= x746x-5 c. f(x)*g(x)- multiply d. f(x)+g(x) (x-1)(x?-5x+4) | Lg )= Papa out 3.5 24dx-x245x-4 if (= not et xx * \g} (x—4)(x—1) 4)(x- 1) x41 fodegtel= x%-6,249%-4 |] £ loge xe rH * Composition of Function (fog)(x)= f(g(x)) (gof)(x)= g(f(x)) Example: 1. f(x) 2(x) x43 2x F(g(x))= (2x0743 g(flx))= 2(x7+3) (fog)(x)= 4x7+3 (gof)(x)= 2x7+6 2. (x) g(x) 4x-5 x44 F(g(x))= 4(x744)-5 =4x7416-5 fog)(x)= 47+11 (F(x))= (4x-5 3744 = 16x7-40x+25+4 (gof)(x)= 16x?-40x+29 3. f(x) g(x) 6x7+2 2x F(g(x))= 6(2x¢7+2