INVERSE FUNCTION // GENERAL MATHEMATICS // TAGALOG

Summary

The video explains how to find the inverse of various mathematical functions step by step. It begins by substituting 'y' for 'f(x)' and then interchanging 'x' and 'y' in the equation. The video demonstrates solving for 'y' through examples involving linear, rational, and cubic functions, and touches on more complex cases like quadratic functions. For each example, the instructor walks through the process of solving the equation, cross-multiplying, simplifying, and factoring to isolate 'y', ultimately determining the inverse function.

Takeaways

• 🔄 The first step in finding an inverse function is to change f(x) to y.
• ↔️ After that, swap x and y in the equation to find the inverse function.
• ➗ Solve for y by isolating it on one side of the equation and simplifying.
• ➕ Example 1: The inverse of f(x) = 3x + 6 is (x - 6) / 3.
• ➕ Example 2: The inverse of f(x) = 2x + 3 is (x - 3) / 2.
• ➕ Example 3: For a more complex function like f(x) = (3x - 7) / (4x + 3), you use cross-multiplication to solve.
• 💡 Use cross-multiplication when the function is a fraction to eliminate denominators.
• 🧮 Factor out common terms when simplifying expressions to find the inverse.
• 🔎 Always check for steps like removing fractions by multiplying both sides by the denominator.
• 🧩 Cube root functions require applying the cube root to both sides to isolate y.

Q & A

What is the first step in finding the inverse function?

The first step is to change f(x) into y, so the equation becomes y = f(x).

After changing f(x) to y, what is the next step?

The next step is to switch the roles of x and y, so x = f(y), and then solve for y.

How do you isolate y in the equation x = 3y + 6?

To isolate y, first subtract 6 from both sides to get x - 6 = 3y. Then, divide both sides by 3 to get y = (x - 6)/3.

What does the inverse function represent in this context?

The inverse function represents the function that 'undoes' the original function, mapping outputs back to inputs.

How do you find the inverse of a function when fractions are involved, such as f(x) = (3x - 7)/(4x + 3)?

Start by switching x and y, then cross-multiply to eliminate the fraction, and solve for y using algebraic methods.

Why is cross-multiplication necessary when solving for the inverse of a rational function?

Cross-multiplication helps eliminate the denominator, simplifying the equation so that you can isolate y and solve for the inverse.

How do you deal with cube roots when finding the inverse of functions like f(x) = x^3 + 5?

To find the inverse, switch x and y to get x = y^3 + 5, then subtract 5 from both sides, and take the cube root of both sides to isolate y.

When dealing with quadratic functions, such as f(x) = x^2 - 2x + 3, what special technique is often used?

Completing the square is a common technique used to simplify quadratic equations when finding their inverses.

What does it mean to 'complete the square' when finding the inverse of a quadratic function?

Completing the square involves rewriting the quadratic equation in the form of (x + a)^2 to simplify solving for y.

Why might some inverse functions include both positive and negative solutions?

Some inverse functions, especially those involving squares or cube roots, can have both positive and negative solutions due to the nature of the operations involved.

Outlines

00:00

🔍 Understanding How to Find the Inverse of a Function

In this section, the process of finding the inverse function is explained. The first step is converting f(x) to y and then swapping x and y. After solving for y, the inverse function formula is derived. The example used involves f(x) = 3x + 6, where the inverse function is determined to be (x - 6) / 3. Another example uses f(x) = 2x + 3, leading to an inverse of (x - 3) / 2.

05:03

➗ Working with Rational Functions to Find Inverses

This section focuses on a more complex example, f(x) = (3x - 7) / (4x + 3), and shows how to find its inverse. The steps involve cross-multiplication, distribution, and combining like terms. After some algebraic manipulation, the inverse function is determined as (3x + 7) / (3 - 4x). Another rational function example, f(x) = (4x + 7) / (2x - 3), follows a similar process, resulting in an inverse function of (-3x - 7) / (2(2 - x)).

10:04

🔄 Finding the Inverse of Cubic and Fractional Functions

This part covers the inverse function for cubic and fractional functions like f(x) = x^3 + 5 and f(x) = (1/2)(3x + 4). The method involves switching x and y, solving for y, and taking cube roots for cubic equations. For fractional functions, fractions are cleared by multiplying both sides by the denominator. The inverse functions are derived as cube root(x - 5) and (2x - 4) / 3, respectively.

15:05

⚖️ Solving Inverse Functions for Cubic Roots and Quadratics

This paragraph demonstrates finding inverse functions for more challenging equations like f(x) = cube root(x - 5) and f(x) = cube root(1 - 4x). The process includes switching x and y, solving by taking cubes, and manipulating the algebra. The results include inverse functions such as (x^3 - 1) / -4. Finally, the quadratic equation f(x) = x^2 - 2x + 3 is introduced, which involves completing the square to find its inverse.

🧮 Applying Completing the Square to Find Inverse of Quadratics

The final section covers a quadratic function, f(x) = x^2 + 2x - 3. The focus is on the technique of completing the square to rewrite the quadratic function in a form that allows solving for y. After performing the algebraic steps, including finding square roots and rearranging terms, the inverse function is found as ±sqrt(x + 2) - 1, highlighting the complexity of finding inverses for quadratic functions.

Keywords

💡Inverse Function

An inverse function is a function that 'reverses' another function, taking the output of the original function and returning the input that produced it. In the context of the video, the inverse function is found by swapping the roles of x and y in the original function and then solving for y. This is a key concept in understanding how functions can be reversed to find their inverses, as demonstrated in the examples provided.

💡Solving for y

In the process of finding an inverse function, 'solving for y' is a step where the original function's equation is manipulated algebraically to express y in terms of x. This is crucial as it sets the stage for swapping x and y to find the inverse. The video script provides several examples of this process, such as solving '3x + 6' for y to get 'y = (x - 6) / 3'.

💡Cross Multiply

Cross multiplication is a technique used in algebra to solve equations involving fractions. In the video, this method is used when dealing with equations like '3x - 7 / 4x + 3'. By cross multiplying, the presenter eliminates the fractions, making it easier to solve for y and subsequently find the inverse function.

💡Cube Root

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In the video, cube roots are used in functions like 'x^3 + 5'. Finding the inverse of such a function involves taking the cube root of the equation, which is shown when the presenter discusses the inverse of 'x^3 + 5' resulting in the cube root of 'x - 5'.

💡Completing the Square

Completing the square is a method used to solve quadratic equations by transforming them into a form that makes them easier to solve. In the video, this technique is applied to quadratic functions to find their inverses. For example, the presenter completes the square for 'x^2 - 2x + 3' to derive its inverse function.

💡Quadratic Function

A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants, and a ≠ 0. The video discusses finding the inverse of a quadratic function, which often involves completing the square or using other algebraic manipulations. The script mentions 'x^2 - 2x + 3' as an example of a quadratic function for which the inverse is derived.

💡Algebraic Manipulation

Algebraic manipulation refers to the process of transforming algebraic expressions through various mathematical operations to achieve a desired form. In the video, algebraic manipulation is used extensively to rearrange functions and solve for the inverse. This includes operations like adding or subtracting terms, factoring, and cross multiplying.

💡Swapping x and y

Swapping x and y is a fundamental step in finding the inverse of a function. By interchanging the roles of x and y in the original function's equation, one can start the process of deriving the inverse function. The video script illustrates this with several examples, such as swapping in 'f(x) = 2x + 3' to get 'x = 2y + 3'.

💡Distributive Property

The distributive property is a fundamental algebraic principle that allows for the multiplication of a term by a sum or difference of other terms. In the video, the presenter uses the distributive property to expand expressions like '4y + 3' when multiplied by 'x', which is necessary for solving equations and finding inverse functions.

💡Factoring

Factoring is the process of breaking down a polynomial or expression into a product of its factors. In the video, factoring is used as part of the process to solve for y and find the inverse function. For example, when dealing with '3y - 7 = 3x + 7', the presenter factors out '3 - 4x' to isolate y and find the inverse.

Highlights

First step in finding the inverse function: change f(x) into y.

After changing f(x) into y, swap x and y in the equation.

Solve for y by isolating it on one side of the equation.

Example 1: For f(x) = 3x + 6, the inverse function is (x - 6) / 3.

For f(x) = 2x + 3, the inverse function is (x - 3) / 2.

For a more complex function like f(x) = (3x - 7) / (4x + 3), cross multiplication is used to find the inverse.

Distribute and combine like terms when solving for y in rational functions.

Factor the left-hand side when isolating y to solve for the inverse.

The inverse of f(x) = (3x - 7) / (4x + 3) is (3x + 7) / (3 - 4x).

For quadratic functions, completing the square may be required to find the inverse.

Handling cube root functions: Solve by isolating the cube root and then taking the cube of both sides.

Example 2: For f(x) = x³ + 5, the inverse function is the cube root of (x - 5).

Example 3: For f(x) = cube root of (1 - 4x), inverse is (x³ - 1) / -4.

Handling square root functions requires squaring both sides of the equation to remove the root.

Inverse functions for quadratics may involve factoring and solving for y with positive and negative square roots.

Transcripts

okay this time guys let us discuss on

how to find the inverse function the

first step that we are going to do

is we have to change this f of x into y

so

i'm giving y then bring down attention

3x plus six

then after that young x governating y

ating y governating x so i'm gonna

give x which is equal to three y plus

six

then after that solve for the value of y

so lipatnates

positive six

x minus six which is equal to three y

then after that angle going out and

solve for the value of y by dividing

both sides by three

by three so therefore this is one so

this is y which is equal to x minus six

all over three so mere nothing inverse

function which is are represented by

this the inverse function

is equal to

x minus six

all over three so i don't know i think

inverse function

supported not an expression i think

final answer guys

step or the the first one on how to find

the inverse function

let us proceed the second one

f of x naught n is equal to two x plus

three

first step and f of x governance uh

shang y

then we have here

bring down two x plus three epoch

palette notice x and y so i'm gigging

x which is equal to 2y plus 3.

then solve for the value of x so region

but not as positive 3 so x minus 3 is

equal to 2y then solve for the value of

y

so divide by dividing both sides by 2 so

2 divided by 2 cancel so this is y which

is equal to

x minus 3

over two

so marinating volume why not in lindito

so therefore the inverse nothing

of a function is

so this is x minus three

over two so this is now the inverse

function at n

okay let us have here another example we

have here f of x is equal to three x

minus seven all over four x plus three

so what we are going to do so bring down

log not n or

which is equal to bring down attention

3x minus 7 all over 4x plus 3.

then you probably don't ncx and y so

i

x which is equal to

three y minus seven all over four y

plus three

then what we're going to do is cross

multiply my imaginary values i determine

a one

so one times 3y minus 7

is just equal to

3y minus 7.

x dominating

multiply number 9

x multiplied by

4y plus three

then distribute netenyon x naught is a

four y plus three so bring down language

three y minus seven

which is equal to shazam for x y so

distribute not n

is three times x is three x

so next logarithmic examination left

side

volume y

so bring down attending

three y so this is three y

[Music]

negative 4 x y

then bring down attention three x so

this is three x

libertas

negative seven magnitude positive seven

then after that factor nothing is a left

side so on gcfnia is y quantity of

um three minus four x

which is equal to three x plus seven

then

after that to get the value of y divide

both sides by

three

minus four x three minus four x and

three minus four x

so cancel

which is equal to

3x plus 7

all over 3 minus 4x

inverse function so which is in the form

of 3x plus 7 all over 3 minus 4x

so this is our inverse function

next manaten

is so we have here f of x is equal to

four x plus seven all over two x minus

three so same as usual f of x

governating y

which is equal to bring down four x plus

seven

all over two x minus three

then epoch particular n c y and x so i'm

going x which is equal to

four y plus seven

all over

two y minus three

then cross multiply not n so my

imaginary value then a one so one times

four y

so this is four y

plus seven

then x naught n multiplied by d so x

naught n multiplying attention

two y minus three

then after that distribute net n bring

down one another four y

plus seven which is equal to x times two

y so two x y

then x times negative three negative

three x

then after that combine like terms latin

value number y

so this is four y bring down attenuation

for y

then it on two x y

is

negative 2xy

which is equal to this is negative 3x

7

is making negative 7

then factor not 10

gcf nothing data is 2y so this is 2y

quantity of

[Music]

2

minus

x which is equal to negative three x

minus seven

so divide nothing both sides by

two two

multiplied by two minus x two minus x

and 2

quantity of 2 minus x

so in this case

with the same thing

is y which is equal to negative 3x

minus 7

all over

2

quantity of

2 minus

x

so this is x

then of course this is now the inverse

function at n young f function nothing

or the inverse function

is equal to

negative 3 x minus 7

all over 2 quantity of 2

minus x so this is now our

inverse function

okay another one so we have here f of x

is equal to one half quantity of four x

plus four

and f of x is equal to x cubed plus five

find the inverse so same as usual this

will become equal to y which is equal to

one half then three x plus four

then after that guys um change the x and

y so it is making x and one half

quantity of three y plus four

then after that uh remove nothing young

fraction at n so but angle at the top

fraction

one half by multiplying by two because

two times one half is equal to one so

multiplying along negative both sides by

two

by two so it is two times x is two x

then

uh two times one half is one

one times three y plus four is three y

plus four so net angle

uh fractional one half so therefore

solve for the value of y natalya so this

is two x then libat

positive 4 negative 4 which is equal to

3y

then solve for the value of y so this is

now uh 3 3.

so

here y value not n is

2x or 2x minus 4 all over 3. so

therefore the inverse function at n

is equal to

x minus 4

all over three

so this is now our

answer

so how about this one a man so f of x is

equal to x cubed plus five so same

routine so this will become y which is

equal to x cubed plus five but palette

now that is x and y so i'm giving x

which is equal to y cubed plus five

then

uh yeah lipid latency positive five mug

again negative five so x minus five is

equal to y cube

so my problem that i did though it is

not a y cube so indeed number nine

putting it divide the n by

cube

okay so angle coming at n is we have to

uh multiply both sides by its cube root

so cube root not n

multiplied nothing by the cube root

so therefore the answer is the cube root

of

x minus 5.

so therefore our inverse function

the inverse function now is not the one

x raised to negative 1

of x is equal to cube root of

x minus five

okay

so this will be your final answer

another example so we have here f of x

is equal to the cube root of x minus

five so same as usual so this is y is

equal to cube root of x minus five

okay

then after that i'll attend x and y so x

is equal to the cube root of y

minus five

then of course um tangalyn cube root by

multiplying by

its cube okay multiply nothing both

sides uh cube multiply nothing both

sides uh cube so it is

x cube which is equal to canceling

attento so this is y

minus five

so

bring down attenuation

x cubed plus five

okay so this is now your inverse

function the inverse function

is now equal to

x cubed plus five

almost the same here sakabila so we have

here f of x is equal to the cube root of

one minus four x so same as usual one

y rather is equal to the cube root of

one minus four x

then interchange so this is x is equal

to the cube root

of one minus four y

then after that multiply both sides by

cube

by cube

so this cube so cancel cancel d is x

cube which is

um

equal to 1 minus 4 y

so this time so we have here

uh positive one magnitude negative one

which is equal to negative four y

since

uh we have here negative four y divide

both sides by negative four negative 4

so this is now equal to

y which is equal to x cubed minus 1 all

over negative 4.

so here our inverse function

is now equal to

x cubed minus 1 all over negative 4.

so this is now our

inverse function

okay so we have here another example f

of x is equal to x squared minus two x

plus three so autonomous in a form of

quadratic then so measure my contain

difference

of quadratic nato so same as usual

so bring down an attention y which is

equal to

x squared plus

two x

minus three

okay so in this case uh

usually

x is equal to

y square

plus

uh

2y

then minus three thousand origin this is

x minus three

tables up four plus three

the opposite is y square

plus

two y

tables

is y quantity of y

plus

two

so it is x plus three suppose divide

both both sides by y plus two y plus two

so on the origin is young y naught

on this kind of quadratic so this time

you apply nothing you're completing the

square or maybe process the entire

parameter

is bring down muna nathan completing the

square taiyo what monetary interchange

number x and y

y

which is equal to

bring down national x squared

then plus 2x

was data but completing the square tire

so completing the square i'm gonna go

nothing

which is one cinerle

one

square nothing

then

negative one square

minus three

so theta simplify monetize y is equal to

x plus one

square

tables

um

negative one square is positive one

then minus three

simplify so this y is equal to

quantity of x plus one

square

one minus three is negative two

so lipo

um negative two

or d times interchange

y so y

plus one

square

minus two

so atom ion

is lipid is negative two

so x

plus two

is equal to

y plus one

so in this case

my squared i d though

so

nothing so by multiplying both sides by

its roots so

so this is a positive negative so

positive negative square root of

x plus two so which is equal to kinase

so this is y

plus one so nipple

your

positive one similarly

negative one

square root of positive negative

na

x plus 2

so this is your y value so inverse

function at n

and inverse function at n

is ethanol which is the negative 1

plus negative square root of

x plus two

okay so this is our inverse function