Menentukan Salah Satu Fungsi Jika Diketahui Fungsi Komposisinya

Summary

The video script is an educational discussion on determining the inverse function given a function's composition. It covers two problems: finding gx when given f(GX) and vice versa. The first problem involves algebraic manipulation to isolate gx, resulting in gx = x^2 - 2x + 2. The second problem involves substituting gx into fx to find the inverse function, leading to a complex algebraic expression for fx. The presenter guides viewers through each step, emphasizing clarity and understanding in mathematical problem-solving.

Takeaways

• 🧑‍🏫 The speaker discusses how to determine a function when its composition with another function is given, using two examples.
• 📝 In the first problem, the given equation is f(g(x)) = 4x² - 8x + 7, with f(x) = 4x - 1. The task is to find g(x).
• 🔄 The process involves substituting g(x) into f(x) and simplifying the expressions to solve for g(x).
• 🧮 The final result for g(x) is x² - 2x + 2 after dividing each term by 4.
• 🧑‍💼 In the second problem, f(g(x)) = x² - 4x + 5 is given, along with g(x) = x + 3. The task is to determine f(x).
• 📊 This time, the process involves substituting g(x) into f(x) and simplifying with a substitution method where x + 3 is replaced by a new variable, t.
• 🧩 After simplification, the function f(t) is expressed in terms of t, and then replaced back with the original variable to find f(x).
• 📐 The final result for f(x) is x² - 10x + 26.
• 🧑‍🔬 The video emphasizes substitution and algebraic manipulation techniques to solve composition problems.
• 🙏 The explanation concludes with thanks and a hope that the audience finds the information beneficial and easy to understand.

Q & A

What is the main topic discussed in the script?

The main topic discussed in the script is the method of determining a function, specifically how to find the function gx given the composition of functions f and g.

What is the given function for f(Bundaran GX) in the first problem?

The given function for f(Bundaran GX) in the first problem is 4x squared minus 8x plus 7.

What is the given function for gx in the first problem?

The given function for gx in the first problem is 4x minus 1.

How does the script suggest to find gx in the first problem?

The script suggests substituting the known function gx into the given function f(Bundaran GX) and solving for gx by isolating it on one side of the equation.

What is the result of the function gx after solving the first problem?

The result of the function gx after solving the first problem is x squared minus 2x + 2.

What is the given function for f(Bundaran GX) in the second problem?

The given function for f(Bundaran GX) in the second problem is x squared minus 4x + 5.

What is the given function for gx in the second problem?

The given function for gx in the second problem is x plus 3.

How does the script suggest to find fx in the second problem?

The script suggests finding the effect (output) of the function by substituting gx into fx and solving for fx.

What is the final expression for fx after solving the second problem?

The final expression for fx after solving the second problem is x squared minus 10x + 26.

What mathematical concept is demonstrated in the script?

The script demonstrates the concept of function composition and solving for an unknown function given the compositions.

What is the significance of the script's conclusion in understanding the process of determining functions?

The conclusion of the script emphasizes the importance of understanding the process of substituting and solving equations to determine unknown functions in function composition.

Outlines

00:00

📘 Determining Function Composition

The paragraph discusses the process of determining the composition of functions given their individual expressions. The speaker explains how to find the function gx when the composition of functions f(G(x)) is known. The example provided involves substituting gx into the function f(x) to derive the expression for gx. The process includes simplifying the expression by moving terms around and isolating the variable x to find the final form of gx. The speaker uses algebraic manipulation to demonstrate how to handle the composition and simplification, resulting in a clear explanation of the function gx.

05:03

🔍 Finding the Effect of a Function Given Its Composition

In this paragraph, the focus is on finding the effect of a function when its composition with another function is known. The speaker provides an example where the composition of functions F(g(x)) is given, and the task is to determine the function f(x). The explanation involves substituting g(x) into F(x) and then simplifying the resulting expression to find f(x). The process includes expanding the terms, combining like terms, and isolating the variable to derive the final form of f(x). The speaker also emphasizes the importance of correctly substituting and simplifying to arrive at the correct function expression, providing a step-by-step guide to understanding function composition and simplification.

Keywords

💡Function Determination

Function determination refers to the process of finding the mathematical expression of a function given certain conditions or compositions. In the video, the theme revolves around this concept, where the presenter discusses how to determine a function 'gx' when the composition of functions 'f' and 'gx' is known. The script uses algebraic manipulation to illustrate this process, such as substituting 'gx' into 'f' to solve for 'gx'.

💡Algebraic Manipulation

Algebraic manipulation is the process of rearranging and combining algebraic expressions to simplify or solve equations. The video script demonstrates this by showing how to manipulate the given function compositions to isolate and solve for the unknown function 'gx'. It is a fundamental concept in mathematics used to understand the relationships between different parts of an equation.

💡Function Composition

Function composition is a concept where one function is applied to the result of another. In the script, the presenter discusses determining 'gx' by knowing the composition of 'f' and 'gx'. This is a key concept in the video, as it is the basis for the algebraic manipulations performed to find the expressions for 'gx' and 'fx'.

💡Substitution

Substitution is a method used in algebra where one expression is replaced with another to simplify equations. In the context of the video, substitution is used to replace 'gx' with 'x' in the function 'f' to find the value of 'gx'. This technique is crucial in solving for the unknown function in the given compositions.

💡Quadratic Function

A quadratic function is a polynomial function of degree two. In the script, the presenter discusses functions involving 'x squared', which is a characteristic of quadratic functions. These functions are essential in the video's examples, as they are the basis for the algebraic expressions being manipulated.

💡Linear Function

A linear function is a polynomial function of degree one. The video script mentions functions with 'x' terms, indicating linear components within the overall algebraic expressions. Understanding linear functions is important for grasping how they contribute to the composition and determination of other functions in the video.

💡Constant Term

A constant term in an algebraic expression is a term that does not contain any variables. In the script, constants like '7' and '1' are part of the function expressions. The presenter shows how these constants are handled during the process of function determination and algebraic manipulation.

💡Coefficient

A coefficient is a numerical factor that multiplies a variable in an algebraic term. The video script discusses coefficients in the context of the 'x' terms, such as '-8x' and '-4x'. Understanding coefficients is essential for correctly determining the function 'gx' and performing the necessary algebraic operations.

💡Simplification

Simplification in algebra refers to the process of making an expression easier to understand or solve by combining like terms or reducing the expression to its simplest form. The video script demonstrates simplification as part of the process to find 'gx' and 'fx', showing how to combine and reduce terms to reach the final function expressions.

💡Inverse Function

An inverse function reverses the effect of the original function, swapping the input and output. In the script, the presenter discusses finding the inverse function 'fx' given the composition of 'f' and 'gx'. This concept is important for understanding the relationship between functions and how they can be manipulated to find one from the other.

💡Variable

A variable in algebra is a symbol, usually a letter, that represents an unknown or changeable quantity. In the video, the variable 'x' is used throughout the script to represent the input of the functions being discussed. Understanding variables is fundamental to grasping the algebraic expressions and operations presented in the video.

Highlights

Introduction to the discussion on determining a function given its composition.

Two problems are presented for explanation.

The first problem involves a function defined as f(Bundaran GX) = 4x^2 - 8x + 7 and FX = 4x - 1.

The task is to find the function GX given the function f(Bundaran GX).

Substitution of known functions into the equation to solve for GX.

Simplification of the equation to isolate GX.

The result shows GX = 4x^2 - 8x + 8 divided by 4.

Final determination of GX as x^2 - 2x + 2.

Introduction to the second problem with a different set of functions.

The second problem involves finding the function FX given f(Bundaran GX) = x^2 - 4x + 5 and GX = x + 3.

Substitution of GX into the function FX to find the effect.

Complex algebraic manipulation to solve for FX.

The process includes squaring, multiplying, and adding constants.

Grouping like terms and constants to simplify the equation.

Final expression for FX is x^2 - 10x + 26.

Conclusion and summary of the method to determine functions given their compositions.

Closing remarks with thanks and a wish for understanding.

End of the discussion with a sign-off in Arabic.

Transcripts

hai hai

Hai

assalamualaikum warahmatullahi

wabarakatuh kali ini saya akan membahas

tentang bagaimana cara menentukan salah

satu fungsi jika diketahui

komposisi fungsinya di sini ada dua soal

yang akan saya bahas Mari kita

selesaikan bersama Bagaimana proses

penyelesaiannya untuk nomor yang pertama

di sini Diketahui

f Bundaran GX = 4 x kuadrat

dikurangi dengan 8X plus dengan 7 dan FX

= 4 x dikurangi dengan satu Tentukan

fungsi gx nya fungsi G adalah berapa di

sini Diketahui f Bundaran

GX kemudian efek dan kita disuruh

menentukan GX yaitu berapa

kita langsung saja bahas penyelesaiannya

bahwa suatu at Bundaran dengar JF

Bundaran GX

= 4 x kuadrat dikurangi dengan 8 x +

dengan 7 f Bundaran GX = f j

tiap pada fungsi gx kita substitusikan

ke dalam fungsi efeknya hasilnya adalah

4 x kuadrat dikurangi dengan 8X ditambah

dengan 7 Padahal di sini GX nya belum

diketahui yang diketahui adalah efeknya

yaitu efeknya 4x dikurangi dengan satu

maka kita langsung saja setiap ada GX

sini kita substitusikan ke dalam fungsi

efeknya yaitu 4x kurangi dengan satu

maka x-nya di sini kita ganti dengan GX

Nya sehingga

4 dikali dengan GX karena kita belum

tahu gxc berapa kemudian dikurangi

dengan satu orang dengan satu Jadi

intinya adalah setiap ada di sini

diganti dengan JX Nya sehingga sini

adalah 4 x kuadrat dikurangi dengan 8X

ditambah dengan tujuh sehingga kita

tujuannya adalah mencari GX setiap angka

atau konstanta di sini kita geser ke

kanan sehingga

4G e4g X

= 4 X kuadrat dikurangi dengan 8X = 7

negatif satunya geser ke ruas kanan

sehingga menjadi positif satu sehingga

4G

X4 DX = 4 x kuadrat dikurangi dengan

8X wujud ditambah 1 hasilnya adalah

delapan kita tujuannya adalah mencari GX

maka GX = 4C ini tidak terus berkarya

maka menjadi Hai sehingga 4 x kuadrat

dikurangi dengan 8 x + dengan delapan

dibagi dengan empat sehingga GX didapat

sama dengan setiap yang pembilang disini

part x kuadrat kurang 8X tambah 8 dibagi

dengan part 4 x kuadrat / 4 hasilnya

adalah x kuadrat atau 1 x kuadrat ya

karena satu tidak perlu dituliskan cukup

dituliskan x kuadrat kemudian negatif 8X

dibagi dengan empat Makasih adalah

negatif

2x 8 dibagi dengan 4 hasilnya adalah

23 rigs nya yaitu X kuadrat dikurangi

dengan 2x + dengan dua itu cara

menentukan fungsi gx nya lanjut untuk

soal yang kedua

untuk yang soalnya dua disini Diketahui

f Bundaran GX = X kuadrat dikurangi

dengan 4x + dengan lima dan fungsi gx

nya adalah x ditambah dengan tiga Kita

disuruh menentukan fungsi fx nya adalah

berapa jadi kebalikan dari tadi kalau

tadi Diketahui f Bundaran GX kemudian

diketahui efeknya kita cari GX kalau ini

diketahui Bundaran GX nya kemudian Desa

diketahui lagi esnya maka kita suruh

mencari efeknya itu berapa ya kita

langsung saja bahas langsung untuk F

Bundaran G Bundaran F Bundaran g x

adalah X kuadrat dikurangi dengan 4x

ditambah dengan 5 f Bundaran GX itu kan

F dalam kurung GX setiap pada fungsi gx

kita subtitusikan kedalam fungsi fx = x

kuadrat yang dikurangi dengan 4x

ditambah dengan lima ini diketahui

fungsi gx adalah x + 3 maka F dalam

kurung x tambah dengan 3 = x kuadrat

dikurangi dengan 4x ditambah dengan lima

kalau tipe soalnya seperti ini

Kita disuruh menentukan efeknya namun

disini adalah F dalam kurung X + 3 maka

yang ada di sini kita misalkan dulu saya

misalkan

a boleh boleh boleh dan seterusnya

asalkan jangan SY di sini ya jangan eh

jangan eh jangan ge saya misalkan disini

tanya ini misalkan teh misal misal

bahwa x tambah dengan 3 = t maka dari

sini kita bisa

eh sama dengan

menjadi kurang Nyi MP3

Astra itu baru kita kembali lagi ke atas

ini

bahwa fx73 kan kita misalkan tadi teh ya

Bapak FT FT = setiap ada di sini kita

ganti dengan teh kurangi saat kurangi 3

Sehingga dalam kurung t uang i3

dikuadratkan

dikurangi dengan empat tali t kurangi

dengan tiga ditambah dengan

5C kurangi tiga kita cari tag kurangi

tiga kuadrat Ya tapi kan artinya C

kurangi tiga dikali dengan teh kurang i3

kita kalikan dulu

TK Lite hasilnya adalah tag wadrat

kemudian Tedi kali dengan negatif 3

adalah negatif 3t

kemudian negatif tiga kali t hasilnya

adalah negatif 3t lagi Kemudian negatif

kali negatif

khas ini adalah flash 9 maka = t kuadrat

negatif negatif 3t kurangi 3tech maka

negatif 6

ditambah dengan 9 maka hasil dari TPU

RAM 3 kuadrat adalah ini ya maka saya

ganti disini FT

= X kurang 3 kuadrat yaitu hasilnya

kuadrat dikurangi dengan 6t ditambah

dengan 9 janin dikalikan dikali depan

kali belakang juga negatif 4 kali t

negatif 4E

kemudian negatif kali negatif hasilnya

adalah plus 12

ditambah dengan Dima Pak

after6 adengan

Nah kita kelompokkan yang punya teh

koefisien variabel dengan variabel

Taeyang konstanta dengan konstanta di

sini Teh syaratnya cuma ada satu maka

kita Tuliskan teks kuadrat kurangi

dengan enam teh kemudian kurangi lagi 4T

udah yang konstanta tambah dengan 9

ditambah dengan

12 tambah dengan

5ac geser kesini ya

geser kesini segini FT

= t kuadrat x kuadrat

kemudian negatif 6tag kurangi 4T jelas

aslinya adalah negatif 10 eh kemudian 9

ditambah dengan 12 itu 21 ditambah

dengan 5 hasilnya adalah 26 plus 26 kita

tujuannya adalah mencari FX disini Bukan

FT maka

efek-efek seperti setiap ada teh disini

diganti dengan

x3x kuadrat dikurangi dengan

10

t-shirt tambah dengan

26

Mohon maaf ini 10x ya ini 10x karena

setiap ada teh kita gantikan dengan x

sehingga fx = x kuadrat dikurangi dengan

10x ditambah dengan

26 itulah cara menentukan efek jika

diketahui F Bundaran GX dan GX ya Terima

kasih semoga bermanfaat dan mudah

dipahami saya kiri wassalamu'alaikum

warahmatullahi wabarakatuh

[Musik]