INVERSE OF ONE-TO-ONE FUNCTIONS || GRADE 11 GENERAL MATHEMATICS Q1

Summary

This video script explains the concept of inverse functions in mathematics. It covers the criteria for a function to have an inverse, including being one-to-one, and demonstrates how to find the inverse of various functions through examples. It also discusses when a function does not have an inverse, such as quadratic functions, and concludes with a problem-solving example converting temperatures between Fahrenheit and Kelvin.

Takeaways

• 🔄 The inverse of a function reverses the input and output, swapping the domain and range of the original function.
• 📉 A function has an inverse if and only if it is one-to-one, meaning each output corresponds to exactly one input.
• 📚 To find the inverse of a function, you interchange x and y, then solve for y in terms of x.
• 📐 The horizontal line test can be used to determine if a function is one-to-one; if any horizontal line intersects the graph more than once, it's not one-to-one.
• 🔢 For a function like y = 2x - 1, substituting values into the function and then reversing the x and y values helps find the inverse.
• ✅ Examples are given to illustrate finding inverses, such as y = 3x + 1 leading to an inverse of y = (x - 1) / 3.
• 📉 Quadratic functions are not one-to-one because their graphs are parabolas that do not pass the horizontal line test.
• 🔢 The absolute value function, like f(x) = |3x|, also fails the horizontal line test and thus does not have an inverse.
• 🌡 Converting temperature scales involves understanding inverse functions, such as converting from Fahrenheit to Kelvin and vice versa.
• ❌ Not all functions have inverses; if a function fails the horizontal line test, it does not have an inverse function.
• 🔄 To check if two functions are inverses, you can use function composition (f(g(x)) should equal x, and g(f(x)) should also equal x).

Q & A

What is the definition of an inverse function?

An inverse function reverses the effect of the original function. If a function f performs a certain operation on x to produce y, then the inverse function undoes that operation to get back to x.

How is the domain and range of a function related to its inverse?

The domain of the inverse function is the range of the original function, and the range of the inverse function is the domain of the original function.

What is the process of finding the inverse of a function?

To find the inverse of a function, you first write the function in the form y = f(x), then interchange x and y, and finally solve for y in terms of x.

What is the condition for a function to have an inverse?

A function has an inverse if and only if it is one-to-one, meaning each input is mapped to exactly one output.

How do you determine if a function is one-to-one?

You can determine if a function is one-to-one by using the horizontal line test. If no horizontal line intersects the graph of the function more than once, then the function is one-to-one.

What is the inverse of the function f(x) = 2x - 1?

The inverse of the function f(x) = 2x - 1 is f^(-1)(x) = (x + 1) / 2.

How do you find the inverse of a function given by a set of ordered pairs?

To find the inverse of a function given by a set of ordered pairs, you swap the x and y values of each pair.

What is the inverse of the function g(x) = x^3 - 2?

The inverse of the function g(x) = x^3 - 2 is g^(-1)(x) = (x + 2)^(1/3).

Why can't a quadratic function have an inverse?

A quadratic function cannot have an inverse because it fails the horizontal line test. A parabola intersects any horizontal line at most once, but an inverse function requires a unique input for each output.

How do you convert a temperature from Kelvin to Fahrenheit using the inverse function?

To convert a temperature from Kelvin to Fahrenheit, you use the inverse function: F = (K - 273.15) * (9/5) + 32.

How can you check if two functions are inverses of each other?

You can check if two functions are inverses of each other by composing them and seeing if the result is the identity function, which leaves the input unchanged.

Outlines

00:00

🔄 Understanding Inverse Functions

This paragraph discusses the concept of inverse functions in mathematics. It explains that an inverse function reverses the effect of the original function, swapping the domain and range. The inverse is only possible for one-to-one functions, where each output is produced by exactly one input. The script uses the example of the function y = 2x - 1, demonstrating how to find its inverse by substituting y for x and solving for the new x, resulting in the inverse function x = (y + 1) / 2. The paragraph emphasizes that the inverse function's domain and range are the range and domain of the original function, respectively.

05:02

📘 Steps to Find the Inverse of a Function

The second paragraph outlines the steps to find the inverse of a given one-to-one function. It starts by expressing the function in the form y = f(x), then interchanges x and y, and finally solves for y in terms of x. The paragraph provides examples, such as finding the inverse of f(x) = 3x + 1, which results in the inverse function f^(-1)(x) = (x - 1) / 3. Another example is given for the function g(x) = x^3 - 2, where the inverse is found by solving for y and taking the cube root, resulting in g^(-1)(x) = cube root of (x + 2). The paragraph also explains that a function has an inverse if and only if it is one-to-one.

10:03

🔢 Solving for Inverses of Rational Functions

This paragraph delves into finding the inverse of more complex functions, such as rational functions. It describes the process of solving for y in terms of x by cross-multiplication and rearranging terms. An example is given for the function f(x) = (2x + 1) / (3x - 4), where the inverse is found by interchanging x and y, and then solving for y to get f^(-1)(x) = (2x + 1) / (3x - 2). The paragraph also addresses why certain functions, like quadratic functions and those involving absolute values, do not have inverses because they fail the horizontal line test, meaning they are not one-to-one functions.

15:04

🌡 Converting Temperature Scales with Inverse Functions

The fourth paragraph applies the concept of inverse functions to convert temperature scales. It explains how to find the inverse function that converts from Kelvin to Fahrenheit, given the original function K(T) = (5/9)*T - 32 + 273.15. The process involves solving for T in terms of K, which results in the inverse function T(K) = 9/5*(K - 273.15) + 32. The paragraph also discusses how to determine if two functions are inverses of each other by using the composition of functions and checking if the result is the identity function.

20:07

📢 Engaging with the Audience

The final paragraph is a call to action for the audience to engage with the content by liking, subscribing, and hitting the bell button to stay updated with the channel's videos. It serves as a conclusion to the video script, encouraging viewers to interact with the channel for more educational content.

Keywords

💡Inverse Function

An inverse function is a mathematical function that 'reverses' another function, mapping the outputs of the original function back to its inputs. In the context of the video, the inverse function is crucial for understanding how to reverse the process of a given function. For instance, if the original function is 'y = 2x - 1', the inverse would allow us to find 'x' when given 'y'.

💡Domain and Range

In mathematics, the domain of a function is the set of all possible input values, while the range is the set of all possible output values. The video explains that the domain of the inverse function is the range of the original function, and vice versa. This concept is fundamental when finding the inverse of a function, as it dictates the possible values the inverse function can take.

💡One-to-One Function

A one-to-one function is a type of function where each output is matched with exactly one input, and vice versa. This property is essential for a function to have an inverse because it ensures that reversing the function will not lead to ambiguities. The video emphasizes that a function has an inverse if and only if it is one-to-one, which is a key condition for the existence of an inverse.

💡Horizontal Line Test

The horizontal line test is a graphical method used to determine if a function is one-to-one. If any horizontal line intersects the graph of the function at more than one point, the function is not one-to-one. The video uses this test to explain why certain functions, like the quadratic function, do not have inverses.

💡Substitution

Substitution is a technique used in algebra to replace a variable or expression with another value or expression. In the video, substitution is used when finding the inverse of a function by replacing 'f(x)' with 'y' and then solving for 'x' in terms of 'y'. This method is crucial for isolating 'x' and expressing the inverse function.

💡Interchange of Variables

Interchanging variables is a step in finding the inverse of a function where 'x' and 'y' are swapped in the equation. This step is demonstrated in the video as part of the process to reverse the original function. By swapping the variables, we can express the inverse function in terms of the original variables.

💡Absolute Value Function

The absolute value function returns the non-negative value of a given number. The video discusses how the graph of the absolute value function, which resembles a 'V' shape, fails the horizontal line test and thus does not have an inverse. This is an example of how the shape of a function's graph can determine its invertibility.

💡Composition of Functions

The composition of functions involves applying one function to the result of another. In the context of inverse functions, if the composition of a function and its inverse results in the identity function, then the two functions are inverses of each other. The video uses this concept to verify whether two functions are inverses by checking if their composition equals the identity function.

💡Identity Function

The identity function is a function that has the unique property of leaving its input unchanged. It is often used as a reference point when discussing inverse functions because the composition of a function and its inverse should result in the identity function. The video mentions the identity function in the context of verifying inverses.

💡Quadratic Function

A quadratic function is a polynomial function of degree two, often graphed as a parabola. The video explains that since the graph of a quadratic function is a parabola that opens upwards or downwards, it fails the horizontal line test and therefore does not have an inverse. This is an important example of a function that is not one-to-one.

Highlights

Definition of inverse function and its relationship with the original function's domain and range.

Explanation of how to find the inverse of a function by swapping domain and range.

Example of finding the inverse function for a given set of ordered pairs.

Process of swapping x and y values to find the inverse function of a one-to-one function.

Detailed steps to find the inverse of a function using algebraic manipulation.

Example of finding the inverse of a linear function (3x + 1).

Example of finding the inverse of a cubic function (x^3 - 2).

Explanation of why a quadratic function cannot have an inverse function.

Example of finding the inverse of a rational function (2x + 1 / 3x - 4).

Explanation of how to handle absolute value functions and their inability to have inverses.

Conversion formula from Fahrenheit to Kelvin and its inverse from Kelvin to Fahrenheit.

How to verify if two functions are inverses of each other using composition.

Example of verifying inverse functions using composition for f(x) = -1/2x and g(x) = -2x.

Importance of one-to-one functions in having valid inverses.

Practical application of inverse functions in temperature conversion.

Transcripts

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function but that means inverse McGrail

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original function Ranger sir inverse you

arrange an original function domain

Chester inverse that and Danielle for

example we have the original function y

is equal to 2x minus 1 so for example my

entire table of values from negative 4

to positive 4 if we substitute negative

force to the original function the value

of y is negative 9 if we substitute

negative 3 to the original function the

values negative 7 ok same as negative 2

negative 1 0 1 2 3 4 so this is not the

body so why so kappa'd kakuni noting an

inverse so I know my IRA so much really

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value of x and you X that will be the

value of y so in a pocket apollon saying

apology Megara how to reverse

the original function okay so let's say

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the original function DT can dominate a

negative 4 negative 3 negative 2

negative 1 0 1 2 3 4

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inverse u negative 4 negative 3 negative

2 negative 1 0

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inverse one-to-one function

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range B then the inverse of F denoted as

f place the negative 1 so this is the

symbol for inverse know it's a function

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defined by function or the inverse

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function has an inverse if and only if

it is one-to-one

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function described by the set of ordered

pairs we have 1 negative 3 2 & 1

3 3 4 5 5 7 so again I'm gonna win

inheritance which the party needs of

each ordered pair the original function

is that so a mulligan inverse function

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function is worth the word okay

to find the inverse of a one-to-one

function given the equation Maritime

Putnam sushant being steps first write

the function in the form y is equal to f

of X next interchange the X and y

variables next and is the last have

solved for Y in terms of X for example

number 1 find inverse of f of X is equal

to 3x plus 1 so that is the original

function first we need to change f of X

into y so muggin y equals 3x plus 1 so

finally turning that into f of x and y

so now it interchange the variable so

you why Kagami nothing X Y or X Gaga we

not in Y so magic in X is equal to 3 y

plus 1

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and not in muscle see why Lee but not in

C+ one D talk so mugging X minus 1

equals 3 y again I seen Assad not in

detail volume of Y so para Makua not in

your body now why we need to divide both

sides by 3 so therefore I'm dividing

both sides by 3 and so 3 y 2 by 3 I'm

not eternal my detour X minus 1 divided

over 3 so that is the value of y

therefore I use in a 10 so dabit morning

why are using ulong so that is now the

inverse of 3x plus 1 okay Lyla getting

it on a simple data pertaining thereto I

inverse so the inverse of X is equal to

X minus 1 over 3 that and any learning

steps may be nominated another example

find the inverse of G of X is equal to X

cubed minus 2 again your example a given

function X cubed minus 3 is a 1 to 1 so

therefore LM not enemy function my

inverse store again and gagawin k-pop

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interchange the variables you wipe up a

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and then solve for y now so our notion

of the steps since why you need a hand

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negative 2 it will become X plus 2 is

equal to Y cube a maritime cube no Marin

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cube Jan so I don't go in

by cubing metal toy t-cubed

inattentiveness a Kabila so mugging cube

root of x plus two is equal to y cube

soda money array so now well are you

three don't sigh mugging expose exponent

okay say you nobody nothing cube boom

shaka billah para my wallet on three so

therefore why now is a cube root of x

plus two penny will enter in parenthesis

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parenthesis cube root of x plus 2 again

but a mossad nut is y so he knew root

not in the Hat no both side the cube

retire so yeah

Moroccan salient 3 so many went along

you why do mean cube root of x plus 2

therefore the inverse of X cubed minus 2

is the cube root of x plus 2 okay

next find inverse of f of X is equal to

2x plus 1 over 3x minus 4 so rational

thought major

hababam solution the Gagarina 10 wooden

steps change nothing young FX naught and

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papa returning x human x naught in d top

operator not in my next step solve for y

in terms of x okay so cross

multiplication 2x times 3y minus 4 it

anion is equal to 2y plus 1 of course it

is distribute not into XD to sell of x

times 3y is 3x y minus x times 4 is 4x

is equal to 2y plus 1 since this is odd

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selling it on 3x why Satan a on 3x y -2

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plus 1 para Mis old nut is CY illa

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divided by 3x salamati Terra - copy the

sign - why / is equal to 2 equals copy

4x plus 1 again since why I am sin

assault nothing at make a summary why

not in a 3x minus 2 he divided not and

both side by 3 X minus 2 so therefore

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2 + ma TT rainylin is why it on a man 4x

plus 1 divide 3x minus 2 so he turning

inverse not n so the inverse of function

2x plus 1 over 3x minus 4 is equal to 2x

plus 1 over 3x minus 4

ok so that will be the function again so

you in person at the end is 4x plus 1

over 3x minus 2 so Shannon game steps

the end ok so meconium inverse next

example

find the inverse of f of X is equal to x

squared plus 4x minus 2 first observe

the given function this function is

quadratic so Hopak quadratic the graph

is parabola pinging you bop opposition's

a one-to-one function using the

horizontal line test

hindi therefore this is a quadratic

function with a graph in the shape of

parabola that opens upward it is not a

one-to-one function as it feels no

result alike there so indeed nothing mr.

Sabzian persica seen is a one-to-one

function

okay so Tinga and observed nuan given

function Peres in dicta in XS ol next

find the inverse of f of X is equal to

the absolute value of 3x again absolute

valley and kappa Greenup not in your

absolute value that is letter V but I'll

be honest shape no graph and it fails

also in horizontal line test so the

graph of f of X is equal to absolute

value 3x is shaped like a B whose vertex

is located at the origin this function

fails the horizontal line test and

therefore has no inverse okay nothing in

my opinion I'll give you one problem

solving okay to convert from degrees

Fahrenheit to Kelvin the function is K

of T is equal to 5 over 9 times t minus

32 plus 273.15 where T is the

temperature in Fahrenheit kelvins the SI

unit of the temperature find inverse

function converting the temperature in

Kelvin to degrees Fahrenheit

garnet Z function okay we need to look

at the inverse okay

right the original function next okay so

I know many re after that

okay McGee kinky nailiang not in shadow

we represent K up tsk no Indian Gilligan

me tetanus why not lagging f of X when I

pretend I'm not in a white Saudi top

palette and anything and K so k minus

200 so it taught the 273.15 D but not in

dawn so mugging k minus 273.15 is equal

to 5 over 9 times t minus 32 next okay

since keno who are not in you inverse

okay you know who are nothing in inverse

so I know I'm money are in a on we need

to divide both side by what we need to

divide both side by seeking over why not

end right by not 5 over 9 okay divide

not in both sides by 5 over 9 CC Tunica

5 over 9 right okay so no money are a

after that you our equations but in

evite nothing and both side by 5 over 9

that will become what okay this one nah

okay so can you know what not in

universe vomit ah ha ha OH

next since quinoa nothing in tea not

only but nothing you - 32 disability

that will become 9 over 5 times K minus

273.15 plus 32 and the value of that is

9 of

five times K minus 273.15 plus 32 and

that is also now the inverse function oh

this original function so I thought

named in the end

no no name Indian sauna how to process

this problem Panama not in Malayalam on

okay

like this example determine whether the

function below are inverses of each

other or not for example my given time

function f of X is equal to negative 1/2

X and G of X is equal to negative 2x

pinocchio not in my element I turned 11

function at oh I inverse a sad-sack we

are going to use composition panel on so

you ex not in detour Papa returned not

innovation and G of X okay oh let say f

of X I'm public user axiom valanor G of

X so the main value no G of X negative 2

X so therefore young X new detour Papa

returned up in a negative 2x so my

concern not a new negative to negative 1

times negative X is positive x so habil

annamund u X nah man

none gee Papa little nothing of value

dang f of X which is negative 1/2 X so

therefore u X naught and Papa returned

nothing of value now f of X now negative

1 of X so Omar consoles in egged C

positive 2 so negative times negative

that is positive x kappa equal answer

good now using the composition

ibiza BN by daddy finish on the two

functions are inverses of each other

dot and I know

Gaga meeting in a composition Parimal a

man and a long functional inversely

lasat Issa and then kept an equal eundel

ah one function a on Ibiza bein in verse

Allah says erisa so salomina to tune and

KO about inverse of one-to-one function

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