Representation of Functions - General Mathematics Reviewer PDF

Representation of Functions General Mathematics reviewer  Relation- may Relationship si x at y (x,y) = (Domain, Range) All functions passes the vertical line test, but only one to one function (injective function) passes both horizontal and vertical line test.  Example 1 Determine if the fo...

Representation of Functions General Mathematics reviewer  Relation- may Relationship si x at y (x,y) = (Domain, Range) All functions passes the vertical line test, but only one to one function (injective function) passes both horizontal and vertical line test.  Example 1 Determine if the following relations represent a function. 1. {(q, 0), (w, 1), (e, 2), (t, 3)} a function and is one to one 2. {(-1, -2), (0, -2), (1, -2), (2, -2)} a function but not one to one 3. {(1, 0), (1,1), (1, 2), (1, -2)} NOT A FUNCTION 4. {(x, 3), (y, 4), (z, 3), (w, 4)} a function but not one to one *Not a function if x values are repeated *Not one to one function if x or y values are repeated  Some Types of Functions 1. Linear Function A function f is a linear function if f(x) = mx + b, where m and b are real numbers, and m and f(x) are not both equal to zero. [ basta pwede mo siya gawing f(x) = mx + b ] e.g. F(x)=3x-2 ; m=3, b=-2 2. Quadratic Function A quadratic function is any equation of the form f(x) = ax2+ bx + c where a, b, and c are real numbers and a≠ 0. highest degree must be 2 e.g. f(x) = x2 + 4x -1 ; a=1, b=4, c=-1 3. Constant Function A linear function f is a constant function if f(x) = mx + b, where m = 0 and b is any real number. Thus, f(x) = b. e.g. f(x) = -2 ; the answer is a constant, no variable seen 4. Identity Function A linear function f is an identity function if f(x) = mx + b, where m = 1 and b = 0. Thus, f(x) = x. e.g. F(x)=3x ; no constant seen 5. Absolute Value Function The function f is an absolute value function if for all real numbers x, f(x) = x, for x ≥ 0 ; –x, for x ≤ 0 e.g. |x+2|=9 ; x+2 = 9 and x+2 = -9 ; x=7 and x=-11 6. Piecewise Function A piecewise function or a compound function is a function defined by multiple sub- functions, where each sub-function applies to a certain interval of the main function's domain. e.g. F(x) = 4x+5 x −5 -6, if x < −5 Exercise B Tell whether the function described in each of the following is a linear function, a constant function, an identity function, an absolute value function, or a piecewise function. Linear: y = mx + b Constant: horizontal line Identity: Straight line that 4. 5. passes through the origin Absolute Value: symmetric about the y –axis Piecewise: different curves/lines will be seen Practice Problem A zumba instructor charges according to the number of participants. If there are 15 participants or below, the instructor charges ₱500.00 for each participant per month. If the number of participants is between 15 and 30, he charges ₱400.00 for each participant per month. If there are 30 participants or more, he charges ₱350.00 for each participant per month. 1. Write the piecewise function that describes what the instructor charges. 2. Graph the function. :b¨e≤ Fẅ ‾ẅFx‘eÇe ≤╫¦ Ť½Ð╦ẅ ½Í Ðĺ ½╘  Definition: An exponential function with base b is a function of the form f(x)= bͯ or y=bͯ, where b > 0, 𝑏𝑏 ≠ 1 EXPONENTIAL FUNCTIONS, EQUATIONS and INEQUALITIES EXAMPLES: 1.4 x-1 =16 x 2. y= 2 x 3. 2 ͯ≥26 How they are different? Definition: An exponential expression is an expression of the form a.b + d, x-c where b > 0, 𝑏𝑏 ≠ 1 Seatwork: Determine whether the given is exponential function ,equation, inequality or none of these. Solve the following: Solve the following: THINK.PAIR.SHARE ẅ ůδ₧ ♪Í ĺ╫ ⅝ Write a closing statement or call-to-action here.  * (baliktaran lng ang graph ni 𝑓𝑓(𝑥𝑥) para kay 𝑓𝑓ˉ¹(𝑥𝑥) * Ang 𝑓𝑓(𝑥𝑥) may 𝑓𝑓ˉ¹(𝑥𝑥) kapag one-to-one. Dapat pasado sa vertical line test at horizontal line test. How do you determine the inverse of a one- to-one function?  Example 1. 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥 + 1 y= 2x + 1 #1. C- change 𝑓𝑓(𝑥𝑥) to y 𝑥𝑥 = 2𝑦𝑦 + 1 #2 S- switch x and y x− 1 = 2𝑦𝑦 #3 S- solve for y (x-1)/2 = 𝑦𝑦 #4 C- change y to 𝑓𝑓(𝑥𝑥) Therefore, the inverse of 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥 + 1 is 𝑓𝑓 ˉ¹(𝑥𝑥) = (x-1)/2 Tandaan si y at x hinde pwede maulit sa one to one function Determine the inverse of the following functions: 1. f(x) = 3x + 2 6. f(x) = (x−2)^3 + 1 2. f(x) = −5x − 7 7. A(x) = 5√2x+11 2𝑥𝑥 3. f(x) = 8. f(x) = 4x/(5−x) 5𝑥𝑥−1 4. f(x) = 2𝑥𝑥 + 5 9. h(x) = (1+2x)/(7+x) 5. h(x) = 5−9x 10. f(x) = 6−10x8x+7 Determine the inverse of each function. (Use the activity sheets below) WHO FOUND IT? 1.f(x)= 5x 2.f(x)= x-6 3.f(x)= 5x+16 4.f(x)= -x + 10 5.f(x)= 3𝑥𝑥 - 4 6. f(x)= 𝑥𝑥 - 8  WHO FOUND IT? Anthony has General Mathematics class in the 4 period. Before he arrives at class, he th realizes he lost his General Math homework. He must have left it somewhere in the school before then. He doesn’t remember where or when he lost it, but surely someone must have found it. He needs to figure it out before class starts. Solve the “clues” on the other page. After solving each clue, find your answer on this page and cross off the corresponding box. When you complete all of the clues, there should only be 3 boxes unchecked. This will solve the mystery.. LOGARITHMIC FUNCTIONS CHRISTE A. ELVINIA Subject Teacher THANK YOU!!!  A log function, equation or inequality dapat may log na makikita An exponential function, equation or inequality dapat may variable na exponent One to one function graph must pass vertical line test and horizontal line test RELATION- relationship of domain and range A function f showed the cost of x for every x that cost 500. f(x) = 500x The value of a function 𝑓𝑓 (𝑥𝑥 ) = 𝑥𝑥 2 at x = 2, ipasok lang si 2 sa x2 para maging 4 𝑓𝑓 ° 𝑔𝑔 of 𝑓𝑓 (𝑥𝑥 ) and 𝑔𝑔(𝑥𝑥 ) papasok lang si 𝑔𝑔(𝑥𝑥 ) at papalitan ang lahat ng x ni 𝑓𝑓 (𝑥𝑥 ) 𝐴𝐴 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛𝑛𝑛 ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒, 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡ℎ𝑒𝑒 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑒𝑒 Estimating 𝑓𝑓(𝑚𝑚 + 𝑛𝑛) where 𝑓𝑓 (𝑥𝑥 ) = 𝑥𝑥 2 + 1, papalitan lng ni (m+n) si x then solve 𝑓𝑓(𝑥𝑥 ) A fraction becomes undefine if the denominator is zero Table of values ng rational function, I substitute lng, dapat equal si f(x) kay x Magchange lng ang inequality kapag mag multiply o mag divide ng negative Pag naghanap ng domain ng rational, cguraduhin hinde mag zero ang denominator tapos ang lahat ng numero kasama na ang simbolo ay ℝ Pag naghanap ng range ng rational, cguraduhin hinde mag zero ang denominator tapos ang lahat ng numero kasama na ang simbolo ay ℝ To find X intercept of a rational function, make the numerator zero Horizontal asymptotes of a rational function, if degree of numerator polynomial < degree of denominator horizontal asymptote is seen at y=0 Zeros of a rational function is the x value that makes numerator zero Steps in getting inverse of a function: C- change f(x) to y, S- switch x to y, S- solve for y, C- change y to f --1(x) Kapag hinanapan ng representation ng inverse ng 𝑓𝑓 (𝑥𝑥 ), hanapin muna ang inverse bago mag substitute ng values Critical values are the numbers that will make the inequality undefine or wrong, for equation, compute for x ℤ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 ℝ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒ℎ𝑖𝑖𝑖𝑖𝑖𝑖, 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝, 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛, 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟, 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 … 𝑒𝑒𝑒𝑒𝑒𝑒 ℚ 𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 Inverse of exponentiation is log To get x intercept set y=0 then solve for x, for y intercept set x=0 then solve for y Para hanapin ang horizontal assymptote, tignan ang exponent’s degree ni Numerator < Denominator, then ha: y = 0; N=D then ha = ratio of leading coefficients, N>D no ha