CALCULUS I

WEEK 2 DAY 2

Informatics

Class A

UISI

SLIDE AND ASSIGNMENT

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OUTLINE

Recap

Equalities

Functions

Domain and Range

Graph

Operations on Functions

Inverse

Trigonometry

Others

Exponential

Logarithm

Number Types

Calculus I 4

Recap of Last Lecture

0 -1 -2 -3 ¾ -½ 3 1 2 4 2 𝜋

i

2i

-i

-2i

Im

Re

1 + 2i

Credit to Mr. Arif

RECAP: INEQUALITIES

1. 1

2< 2

1

4+

1

8𝑡 <

3

4

2. −8 < 2 3 + 4𝑥 − 4(1 + 3𝑥)≤3

3. 𝑡2 ≤ 𝑡

4. 𝑧6 + 8𝑧5 + 12𝑧4 ≥ 0

5. 𝑥2+8𝑥+16

𝑥> 0

6. 𝑢−8

3𝑢4−𝑢5 ≤ 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. |𝑢2 − 7𝑢| = 12

2. |𝑧2 − 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 = 𝑥2 + 3 then

• 𝑓 2 = 22 + 3 = 7

• 𝑓 𝑡 = 𝑡2 + 3

• 𝑓 𝑡 + ℎ = (𝑡 + ℎ)2+3 = 𝑡2 + 2𝑡ℎ + ℎ2 + 3

EXERCISES

1. 𝑓 𝑥 = 𝑥2 + 3𝑥 + 6

a. 𝑓 −1

b. 𝑓(𝑥 − 1)

c. 𝑓(𝑥2)

2. 𝑔 𝑡 = 2 − 𝑡2

a. 𝑔 1

b. 𝑔(𝑥

𝑡)

c. 𝑔( 𝑥)

3. ℎ 𝑤 =𝑤

3−𝑤2

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 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 = 𝑥2 + 3𝑥 + 6

2. 𝑔 𝑡 = 2 − 𝑡2

3. ℎ 𝑤 =𝑤

3−𝑤2

4. 𝑓 ℎ = 6

5. 𝑔 𝑡 =𝑡−1

𝑡2−1

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 = 𝑥𝑎, dimana 𝑎 konstanta,

maka disebut fungsi berpangkat.

FUNGSI POLINOMIAL

Suatu fungsi 𝑝 disebut fungsi polinomial jika

𝑝 𝑥 = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + ⋯ + 𝑎1𝑥 + 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 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, disebut fungsi kuadratik.

Sedangkan fungsi kubik dituliskan dengan

polinomial berderajat 3 berikut :

𝑝 𝑥 = 𝑎𝑥3 + 𝑏𝑥2 + 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 = 𝑎𝑥, dimana

𝑎 suatu konstanta positif dan 𝑎 ≠ 1 disebut

fungsi eksponensial.

Fungsi eksponensial memiliki domain −∞, ∞

dan range 0, ∞ .

FUNGSI LOGARITMA

Fungsi 𝑓 𝑥 = log𝑎 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

𝑔𝑥 =

𝑓(𝑥)

𝑔(𝑥)

• 𝑓𝑛 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 = 𝑥2 − 14𝑥

3. 𝑓 𝑥 = 𝑥2 − 2𝑥 + 1, 𝑔 𝑥 = 8 − 3𝑥2

4. 𝑓 𝑥 = 𝑥2 + 3, 𝑔 𝑥 = 5 + 𝑥2

5. 𝑓 𝑠 = 𝑠2 − 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 = 𝑥2 − 3𝑥

5. 𝑓 𝑡 =4

3−𝑡

6. 𝐺 𝑡 =2

𝑡2−16