Informatics Class A UISI CALCULUS I WEEK 2 DAY 2

Informatics Class A UISI CALCULUS I WEEK 2 DAY 2

SLIDE AND ASSIGNMENT teachingnation.wordpress.com

OUTLINE Recap Equalities Functions Domain and Range Graph Operations on Functions Inverse Trigonometry Others Exponential Logarithm

Recap of Last Lecture Number Types Im 2i 1 + 2i i 0-3 -2-1 -½ ¾ 1 2 2 3 π 4 Re -i -2i Calculus I 4 Credit to Mr. Arif

RECAP: INEQUALITIES 1. 1 2 < 2 1 4 + 1 8 t < 3 4 2. 8 < 2 3 + 4x 4(1 + 3x) 3 3. t 2 t 4. z 6 + 8z 5 + 12z 4 0 5. x2 +8x+16 x > 0 6. u 8 3u 4 u 5 0 7. Ms. Bella is planning a rectangular garden that is to be twice as long as its wide. If she can afford to buy at most 180 feet of fencing, then what are the possible values for the width?

8. A pilot started at point A and flew in the direction shown in the diagram for some time. At point B she made a 110º turn to end up at point C, due east of where she started. If the measure of angle C is less than 85º, then what are the possible values for x? Exercises are taken from: - tutorial.math.lamar.edu - http://www.mhhe.com/math/devmath/dugopolski/elem/student/olc/graphics/dugopolski03ea_s/ch02/others/ch02-9.pdf

RECAP: ABSOLUTE VALUES 1. u 2 7u = 12 2. z 2 6 = z 3. The velocity of a failed rocket can be modeled with the function v(t) = -20t+140, where t means time in seconds, and v is measured in meters/second. At what points in time is the magnitude of the velocity greater than 100m/s? 4. 5. 6.

Functions

FUNCTIONS Functions are a tool for describing the real world in mathematical terms. The temperature at which water boils depends on the elevation above sea level The interest paid on a cash investment depends on the length of time the investment is held. The area of a circle depends on the radius of the circle the value of one variable quantity, say y, depends on the value of another variable quantity, which we might call x.

y = ƒ(x) ( y equals ƒ of x ) the letter x is the independent variable representing the input value of ƒ, and y is the dependent variable or output value of ƒ at x.

A single letter like f or F is often used to name a function f(x) for example, denotes the value that function f assigns to x Example: If f x = x 2 + 3 then f 2 = 2 2 + 3 = 7 f t = t 2 + 3 f t + h = (t + h) 2 +3 = t 2 + 2th + h 2 + 3

EXERCISES 1. f x = x 2 + 3x + 6 a. f 1 b. f(x 1) c. f(x 2 ) 2. g t = 2 t 2 a. g 1 b. g( x t ) c. g( x) 3. h w = w 3 w 2 a. h 1 b. h(x + t) c. h( 1 t + x)

DOMAIN AND RANGE The set D of all possible input values is called the domain of the function. The set of all values of ƒ(x) as x varies throughout D is called the range of the function.

DOMAIN AND RANGE

GRAPH OF FUNCTIONS

THE VERTICAL LINE TEST FOR A FUNCTION Not every curve in the coordinate plane can be the graph of a function. A function ƒ can have only one value for each x in its domain, so no vertical line can intersect the graph of a function more than once. If a is in the domain of the function ƒ, then the vertical line will intersect the graph of ƒ at the single point (a, ƒ(a))

INCREASING AND DECREASING FUNCTION If the graph of a function climbs or rises as you move from left to right, we say that the function is increasing. If the graph descends or falls as you move from left to right, the function is decreasing.

FUNGSI NAIK DAN FUNGSI TURUN Definisi : Misalkan f fungsi yang didefinisikan pada interval I, dan misalkan x 1 dan x 2 titik-titik pada interval I. 1. Jika f(x 2 ) > f(x 1 ) saat x 1 < x 2, maka f disebut fungsi naik di I. 2. Jika f(x 2 ) < f(x 1 ) saat x 1 < x 2, maka f disebut fungsi turun di I.

The function graphed in Figure 1.9 is decreasing on [-, 0]and increasing on [0, 1]. The function is neither increasing nor decreasing on the interval [1, ] because of the strict inequalities used to compare the function values in the definitions.

FUNGSI GENAP DAN FUNGSI GANJIL Definisi : Fungsi y = f(x) adalah 1. Fungsi genap, jika berlaku f x = f x, 2. Fungsi ganjil, jika berlaku f x = f x, untuk setiap x merupakan domain fungsi. Catatan : - Grafik fungsi genap simetris dengan sumbu y - Grafik fungsi ganjil simetris dengan titik pusat.

CONTOH :

Determine whether the following functions are odd, even, or neither 1. f x = x 2 + 3x + 6 2. g t = 2 t 2 3. h w = w 3 w 2 4. f h = 6 5. g t = t 1 t 2 1

FUNGSI LINEAR Suatu fungsi berbentuk f x = mx + b, untuk setiap konstanta m dan b disebut fungsi linear. Gambar (a) menunjukkan garis f x = mx, dimana b = 0. Sehingga garis yang terbentuk selalu melalui origin (titik pusat). Suatu fungsi f x = x, dimana m = 1 dan b = 0 disebut fungsi identitas.

FUNGSI BERPANGKAT Suatu fungsi f x = x a, dimana a konstanta, maka disebut fungsi berpangkat.

FUNGSI POLINOMIAL Suatu fungsi p disebut fungsi polinomial jika p x = a n x n + a n 1 x n 1 + + a 1 x + a 0 dimana n bilangan bulat nonnegatif dan a 0, a 1,, a n konstanta real (yang disebut koefisien polinomial). Semua polinomial memiliki domain di (, )

FUNGSI POLINOMIAL Fungsi Linear dengan m 0 adalah polinomial berderajat 1. Polinomial berderajat 2, dituliskan sebagai p x = ax 2 + bx + c, disebut fungsi kuadratik. Sedangkan fungsi kubik dituliskan dengan polinomial berderajat 3 berikut : p x = ax 3 + bx 2 + cx + d

FUNGSI POLINOMIAL

FUNGSI RASIONAL Suatu fungsi rasional adalah hasil bagi/ perbandingan f x = p(x)/q(x), dimana p dan q polinomial.

FUNGSI TRIGONOMETRI

FUNGSI EKSPONENSIAL Suatu fungsi yang berbentuk f x = a x, dimana a suatu konstanta positif dan a 1 disebut fungsi eksponensial. Fungsi eksponensial memiliki domain, dan range 0,.

FUNGSI LOGARITMA Fungsi f x positif. = log a x, dengan a 1 suatu konstanta

OPERATIONS ON FUNCTIONS f + g x = f x + g x f g x = f x g(x) f g x = f(x) g(x) f g f n x x = f(x) g(x) = [f x ] n f g x = f(g(x))

FUNGSI KOMPOSISI Definisi : Jika f dan g suatu fungsi, maka komposisi fungsi f g (f komposisi g) didefinisikan sebagai f g x = f g x. Domain dari f g terdiri dari bilangan x pada domain g yang terletak pada domain f.

FUNGSI KOMPOSISI Contoh 2 : Jika f x = x dan g x = x + 1, hitung (a) f g x (c) f f x (b) g f x (d) g g x

Compute f g and g f for each of the following pairs of functions 1. f x = 4x 1, g x = 6 + 7x 2. f x = 5x + 2, g x = x 2 14x 3. f x = x 2 2x + 1, g x = 8 3x 2 4. f x = x 2 + 3, g x = 5 + x 2 5. f s = s 2 4, g w = 1 + w

SOAL LATIHAN Temukan domain dan range pada masing-masing fungsi berikut : 1. f x = 1 + x 2 2. f x = 1 x 3. F x = 5x + 10 4. g x = x 2 3x 5. f t = 4 3 t 6. G t = 2 t 2 16