Informatics Class A
UISI
CALCULUS I WEEK 2 DAY 2
SLIDE AND ASSIGNMENT
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OUTLINE Recap Equalities Functions
Domain and Range Graph Operations on Functions Inverse Trigonometry
Others Exponential Logarithm
Recap of Last Lecture
Number Types Im 1 + 2i 2i
i
-3
-2
-1
0
-½
Re ¾ 1
2
2
3𝜋
4
-i
-2i
Calculus I
Credit to Mr. Arif
4
RECAP: INEQUALITIES 1 2
1 4
1 8
3 4
1. < 2
2. −8 < 2 3 + 4𝑥 − 4(1 + 3𝑥)≤3
3. 𝑡 2 ≤ 𝑡
4. 𝑧 6 + 8𝑧 5 + 12𝑧 4 ≥ 0
+ 𝑡 <
5.
𝑥 2 +8𝑥+16 𝑥
6.
𝑢−8 3𝑢4 −𝑢5
7. Ms. Bella is planning a rectangular garden that is to be twice
>0
≤0
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. |𝑢2 − 7𝑢| = 12 2 2. |𝑧 − 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 = 22 + 3 = 7 • 𝑓 𝑡 = 𝑡2 + 3 • 𝑓 𝑡 + ℎ = (𝑡 + ℎ)2 +3 = 𝑡 2 + 2𝑡ℎ + ℎ2 + 3
EXERCISES 1.
2.
3.
𝑓 𝑥 = 𝑥 2 + 3𝑥 + 6 a.
𝑓 −1
b.
𝑓(𝑥 − 1)
c.
𝑓(𝑥 2 )
𝑔 𝑡 = 2 − 𝑡2 a.
𝑔 1
b.
𝑔( 𝑡 )
c.
𝑔( 𝑥)
𝑥
ℎ 𝑤 =
𝑤 3−𝑤 2
a.
ℎ 1
b.
ℎ(𝑥 + 𝑡)
c.
ℎ( + 𝑥)
1 𝑡
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 𝑥1 dan 𝑥2 titik-titik pada interval 𝐼. 1. Jika 𝑓(𝑥2 ) > 𝑓(𝑥1 ) saat 𝑥1 < 𝑥2 , maka 𝑓 disebut fungsi naik di 𝐼. 2. Jika 𝑓(𝑥2 ) < 𝑓(𝑥1 ) saat 𝑥1 < 𝑥2 , 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 − 𝑡2
3.
ℎ 𝑤 =
4.
𝑓 ℎ =6
5.
𝑔 𝑡 =
𝑤 3−𝑤 2
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 = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0 dimana 𝑛 bilangan bulat nonnegatif dan 𝑎0 , 𝑎1 , … , 𝑎𝑛 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 = log 𝑎 𝑥, dengan 𝑎 ≠ 1 suatu konstanta positif.
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
𝑥 =
𝑓(𝑥) 𝑔(𝑥)
• 𝑓 𝑛 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 + 𝑥2 2. 𝑓 𝑥 =1− 𝑥 3.
𝐹 𝑥 = 5𝑥 + 10
4.
𝑔 𝑥 = 𝑥 2 − 3𝑥
5.
𝑓 𝑡 =
6.
𝐺 𝑡
4 3−𝑡 2 = 2 𝑡 −16
