Lec 44 - One-to-One Function: Examples & Theorems

Summary

This video script explores the concept of one-to-one functions, providing examples to illustrate the concept. It begins by examining the modulus function, demonstrating it is not one-to-one using the horizontal line test. It then contrasts this with the linear function, f(x) = x, which is shown to be one-to-one. The script further delves into cubic functions, proving their one-to-one nature through the same test. The presenter also introduces the theorem that a function is one-to-one if no horizontal line intersects its graph more than once. Finally, the script identifies increasing and decreasing functions as classes of one-to-one functions, concluding with the notion that one-to-one functions are common and reversible, setting the stage for a discussion on exponential functions.

Takeaways

• 📚 The script discusses the concept of one-to-one functions, also known as injective functions, in the context of mathematics.
• 🔍 It uses the modulus function, f(x) = |x|, as an example to demonstrate that it is not a one-to-one function due to the same output for different inputs (e.g., f(2) = f(-2) = 2).
• 📈 The script explains the vertical line test to determine if a relation is a function, ensuring no vertical line intersects the graph more than once.
• 📉 The horizontal line test is introduced as a method to check if a function is one-to-one, where a horizontal line should intersect the graph at most once.
• 👍 The linear function f(x) = x is confirmed to be one-to-one because for any two different inputs x1 and x2, the outputs f(x1) and f(x2) are also different.
• 📊 The script suggests that cubic functions, represented graphically with an example, are one-to-one due to their nature of not intersecting with a horizontal line at more than one point.
• 📘 A theorem is presented: if any horizontal line intersects the graph of a function at most once, the function is one-to-one.
• 🔑 The proof of the theorem is outlined, stating that if a function is not one-to-one, a horizontal line will intersect its graph at more than one point.
• 📝 The class of functions that are one-to-one is explored, highlighting that strictly increasing or decreasing functions are one-to-one based on the horizontal line test.
• 📉 The script provides intuition for why increasing and decreasing functions are one-to-one, as they satisfy the condition that for any x1 < x2, f(x1) < f(x2) for increasing functions and f(x1) > f(x2) for decreasing functions.
• 🌐 The script concludes by emphasizing that one-to-one functions are common and can be reversible, setting the stage for further topics such as exponential functions.

Q & A

What is a one-to-one function?

A one-to-one function, also known as an injective function, is a function where each input is mapped to a unique output, meaning no two different inputs will have the same output.

How can you determine if the function f(x) = |x| is one-to-one?

You can determine if the function f(x) = |x| is one-to-one by using the horizontal line test. If no horizontal line intersects the graph of the function at more than one point, then the function is one-to-one. In the case of f(x) = |x|, it is not one-to-one because, for example, f(2) = f(-2) = 2.

What is the vertical line test, and how does it apply to the function f(x) = x?

The vertical line test is a graphical method to determine if a curve is the graph of a function. If every vertical line intersects the curve at most once, then it passes the vertical line test and is a valid function. For f(x) = x, the vertical line test is passed because each vertical line intersects the line y = x at exactly one point.

Why is the function f(x) = x considered one-to-one?

The function f(x) = x is one-to-one because for any two different inputs x1 and x2, the outputs will also be different (f(x1) = x1 and f(x2) = x2), meaning x1 ≠ x2 implies f(x1) ≠ f(x2).

What is the horizontal line test, and how does it relate to one-to-one functions?

The horizontal line test involves drawing horizontal lines across the graph of a function and checking if any line intersects the graph at more than one point. If no horizontal line intersects the graph more than once, the function is one-to-one.

Can cubic functions be one-to-one? If so, under what conditions?

Cubic functions can be one-to-one depending on their specific form and coefficients. A cubic function is one-to-one if it is strictly increasing or strictly decreasing, meaning that as x increases, f(x) also increases or decreases without any reversals.

What is the theorem stated in the script regarding the horizontal line test and one-to-one functions?

The theorem states that if any horizontal line intersects the graph of a function at most in one point, then the function is one-to-one.

How can you prove that if a function is not one-to-one, it will intersect some horizontal line at more than one point?

If a function is not one-to-one, there exist at least two different inputs x1 and x2 that yield the same output (f(x1) = f(x2)). By plotting these points on the graph and drawing a horizontal line at the level of the common output, this line will intersect the graph at both points (x1, f(x1)) and (x2, f(x2)), demonstrating the function is not one-to-one.

What class of functions are generally one-to-one?

Increasing functions and decreasing functions are generally one-to-one. An increasing function is one where if x1 < x2, then f(x1) ≤ f(x2), and a decreasing function is one where if x1 < x2, then f(x1) ≥ f(x2).

Why are one-to-one functions considered abundant in nature and reversible?

One-to-one functions are abundant because they represent a wide variety of mathematical relationships where each input corresponds to a unique output. They are reversible because knowing the output allows you to uniquely determine the input, which is a desirable property in many mathematical and real-world applications.

What is the next topic discussed in the script after one-to-one functions?

The next topic discussed in the script is exponential functions.

Outlines

00:00

📚 Understanding One-to-One Functions

The script begins by introducing the concept of one-to-one functions, using the modulus function as an example. It explains that the modulus function, which equals x for non-negative values and -x for negative values, forms a V-shape graph and is not one-to-one due to the same output value for both x and -x. The script then contrasts this with the linear function f(x) = x, which is proven to be one-to-one by the horizontal line test, ensuring no two different inputs result in the same output. The discussion emphasizes the importance of the vertical and horizontal line tests in determining the validity and one-to-one nature of functions.

05:07

📉 Analyzing the One-to-One Property in Cubic Functions

The second paragraph delves into the analysis of cubic functions, aiming to determine if they are one-to-one. The script suggests that cubic functions, due to their nature of curving upwards or downwards, will pass the horizontal line test, indicating that for any two distinct inputs, x1 and x2, the outputs f(x1) and f(x2) will also be distinct. This property is confirmed by the theorem that if a horizontal line intersects the graph of a function at most once, the function is one-to-one. The script also touches on the graphical proof of this theorem, demonstrating that non-one-to-one functions will intersect a horizontal line at more than one point.

10:13

🔍 Identifying Classes of One-to-One Functions

The final paragraph focuses on identifying classes of functions that are inherently one-to-one. It discusses the characteristics of increasing and decreasing functions, explaining that if for any two inputs x1 and x2, the function satisfies the condition that f(x1) < f(x2) when x1 < x2 (for increasing functions) or f(x1) > f(x2) when x1 < x2 (for decreasing functions), then the function is one-to-one. The script uses the horizontal line test to illustrate that both increasing and decreasing functions will pass this test, confirming their one-to-one nature. The summary concludes by noting that one-to-one functions are common and often reversible, setting the stage for the discussion of exponential functions in the subsequent topic.

Keywords

💡One-to-One Function

A one-to-one function, also known as an injective function, is a type of mathematical function where each input is mapped to a unique output, ensuring that no two different inputs will produce the same output. In the video, the concept is explored through various examples to demonstrate when a function is not one-to-one and when it is, using the horizontal line test as a method to determine injectivity.

💡Modulus Function

The modulus function, denoted as |x|, is a mathematical function that returns the absolute value of x, which is the non-negative value of x regardless of its sign. In the script, the modulus function is used as an example to illustrate a function that is not one-to-one because |2| equals |-2|, showing that two different inputs can lead to the same output.

💡Vertical Line Test

The vertical line test is a graphical method used to determine if a curve is the graph of a function. If any vertical line intersects the curve in more than one point, then the curve does not represent a function because a function must pass the vertical line test, having each input value correspond to no more than one output value. The video script uses this test to confirm whether certain graphs represent valid functions.

💡Horizontal Line Test

The horizontal line test is another graphical method used to determine if a function is one-to-one. If a horizontal line intersects the graph of a function at more than one point, then the function is not one-to-one. The script explains this concept and uses it to demonstrate that certain functions, such as linear and cubic functions, are one-to-one because no two different inputs produce the same output.

💡Linear Function

A linear function is a function of the form f(x) = mx + b, where m and b are constants, and m is not equal to zero. It represents a straight line when graphed. In the video, the linear function f(x) = x is used as an example of a one-to-one function because it passes the horizontal line test, ensuring that each x-value corresponds to a unique y-value.

💡Cubic Function

A cubic function is a polynomial function of degree three, typically represented as f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants, and a is not zero. The script mentions cubic functions and uses them to illustrate that they can also be one-to-one, depending on their specific form and the domain considered.

💡Injective

Injective is an adjective used to describe a function that is one-to-one. It comes from the Latin word 'injicere', meaning 'to throw into'. In the context of the video, a function is injective if it passes the horizontal line test, meaning each input is mapped to a unique output with no duplicates.

💡Increasing Function

An increasing function is a type of function where the output values increase as the input values increase. If x1 < x2, then f(x1) ≤ f(x2). The video script explains that increasing functions are one-to-one because they pass the horizontal line test, ensuring that no two different inputs result in the same output.

💡Decreasing Function

A decreasing function is the opposite of an increasing function, where the output values decrease as the input values increase. If x1 < x2, then f(x1) ≥ f(x2). The script uses decreasing functions to illustrate that they too are one-to-one, as they satisfy the conditions of the horizontal line test.

💡Exponential Function

An exponential function is a mathematical function of the form f(x) = a * b^x, where a and b are constants, and b > 0 and b ≠ 1. The script mentions exponential functions towards the end, hinting at their exploration in the context of one-to-one functions, although the specific details are not elaborated within the provided transcript.

Highlights

Introduction to the concept of one-to-one functions and the criteria for identifying them.

Explanation of the modulus function and its graphical representation as a V-shape.

Verification of the modulus function as a valid function using the vertical line test.

Determination that the modulus function is not one-to-one by comparing f(x) values for x=2 and x=-2.

Introduction of the linear function f(x) = x and its graphical representation as a straight line.

Confirmation of the linear function as a valid function and its one-to-one property using the horizontal line test.

Theorem statement: A function is one-to-one if a horizontal line intersects its graph at most at one point.

Graphical proof of the theorem by showing a non-one-to-one function intersecting a horizontal line at two points.

Identification of increasing functions as a class of one-to-one functions.

Graphical demonstration of an increasing function satisfying the horizontal line test.

Identification of decreasing functions as another class of one-to-one functions.

Graphical representation of a decreasing function and its one-to-one property.

Discussion on the abundance and reversibility of one-to-one functions in nature.

Introduction to exponential functions as the next topic of discussion.

Transcripts

So, let discuss some examples of a functions that are one to one and not one to one . So,

for this let us first take f x to be equal to modulus of x; is this function one to one?

Try let us try to answer this question. So, let me write this function properly. So, if

f of x is equal to x for x greater than or equal to 0 and minus x for x less than 0 . So,

it is actually a straight line on a passing through the origin like this 40 at a 45 degree

angle and the minus x is this line ok. So, so, so it is a V shape 90 degrees V hm; so,

is this function one to one. First of all let us let us not take the argument, first

of all is this a function f of x is equal to mod x?

Pass a vertical line, take a vertical line and pass it through this; is there at if there

is any point where two points more than one points pass through this function pass through

that line then it is not a function. So, vertical line test is successful therefore, it is a

function. Vertical line test says that it is a function

succeeds and we know it is a function ok. Now, the question is is the function one to

one? Right. So, you pass a horizontal line. So, let me pass one horizontal line somewhere,

let us take this horizontal line. Now is the function one to one?

For x 1 not equal to x 2, I got the same f of x ok. So, how will I prove it is not one

to one? Let us take avalue which is say 2 and minus 2; these are the two values, f of

2 is equal to 2 and f of which is also equal to f of minus 2 ok. Therefore, this is not

going to be a one to one function. So, it is not one to one function . So, our conclusion

is it is not one to one function . Then do we know functions that are one to one?

So,since we have taken f of x is equal to mod x, let us take a function f of x to be

equal to x; is this function one to one? It is a straight line passing through the origin,

is this function one to one? Let us take horizontal , firstlet us check whether this is a function,

take a horizontal line, pass it through thispass it horizontally, the line parallel to x axis.

So, just drag x axis up and down; do you see any point touching more than one point? No.

So, it is a valid function; then, so, sorryyeah you have to pass the vertical line first ok.

Start with f of x is equal to x, take a vertical line which is y axis, slide it to the left,

slide it to the right. Do you see any where it has more than one points? No. So, it is

a valid function. Then take a horizontal line, pass it from the top to bottom; see if if

you are getting any any two points together for on that line; no.

Therefore, this function is actually one to one because x 1 not equal to x 2 implies f

of x 1 is not equal to f of x 2 which is more or less expected right. Because f of x is

equal to x therefore, x 1 not equal to x 2 will give x 1 not equal to x 2, that is f

of x 1 not equal to f of x 2 . So, what about it is an exercise then what

about if you take a cubic functions? So, cubic function will pass like this sorry,it is not

a correct diagram of a cubic function. So, cubic function let us change the color as

well , cubic function will have something like this , symmetry will be retained and

then this will go down. So, if this function, now check whether this

function is one to one or not . Again the exercise is very similar, pass a let the x

axis go up and down, see if you are finding any two points together. So, let us say this

function is f of x is equal to x cube hm and now you can easily make out that for x 1 not

equal to x 2 f of x 1 is not equal to f of x 2 . So, again through horizontal line test,

I have detected that the function is one to one .

So, let us write this particular test as a theorem. If any horizontal line intersects

the graph of a function in at most one point, then the function is one to one ok. So, then

what we will show here,if you want the proof of this what we will show here is if the function

is not one to one then it will intersect some horizontal line will intersect the graph of

a function in more than one point ok. So, that is very easy to prove . So, I I will

prove it graphically. So, if the function is not one to one, let

us say this is x axis, this is y axis. If the function is not one to one, I can take

this point and call this as x 1 and I can take this point as call this as x 2. This

is how I can make function not one to one and then pass a curve passing through these

two points and pass the horizontal line over here which we have done several times now

by now . And therefore, f of x 1 and f of x 2 are same

, they both are same. Therefore, the function is not one to one, that essentially proves

the point that if a horizontal line intersects the graph of a function in at most one point

then f is one to one good. So, we are good to go now. Next thing that

we will come is can we identify the class of functions that are one to one? So, what

class of functions can you immediately think are one to one? For example, we have also

seen some functions like if x 1 is less than or equal to x 2 then f of x 1 is less than

or equal to f of x 2 hm or let us not put this strict equality; let us put this way

strictly increasing. So, what what does what do I mean by ok; let

uscan wequestion is can we identify the class of functions that are not one to one? So,

I I can; obviously, think of function of this form x 1 less than x 2 f of x 1 less than

f of x 2 . Let me plot itand then the my imagination will work fine. So, this function is something

like if x 1 is to the left of x 2 then f of x 1 should always be lessto the left of f

of x 2. Or, if you are plotting it on the y axis then

f of x 1 should be below f of x 2, this is the intuition and you can draw line joining

these two points. Let it go ahead and this is true for every x 1 x for every x 1, x 2

belonging to A this is true; then I am done. But, this function have a name that is they

are called increasing functions ok. In a similar manner, if I multiply this this

function with minus sign. Then I will get a function which is decreasing function and

that can be written as x 1 less than x 2 employs f of x 1 greater than f of x 2 and this is

called decreasing function. Now, you look at any increasing function and

apply your horizontal line test. What is the horizontal line test? Just now we have seen

that it theif you take the horizontal line, roll it across the axis across y axis and

there should not be more than one point intersecting that line at any given point in time ok. So,

this increasing function and decreasing function will satisfy this phenomena.

And therefore, we can easily write this as through verticalthrough horizontal line test

that, if f is an increasing function or a decreasing function then f is one to one.

Let us see one decreasing function as well. What happens when the function is decreasing?

As I go from left to right there is a x 1 is here, x 2 is here . As I go from left to

right, I get x 1 here and now according to the condition f of x 1 should be greater than

f of x 2. So, it will be somewhere here and I can have a line passing through not line,

but the curve passing through this point in this manner ok.

This is true for every x 1 and x 2 belonging to the domain. And therefore, using our line

test, horizontal line test we can easily see that the function whether it is increasing

or decreasing , they are one to one. This gives us a big class of functions that are

one to one . So, one to one functions are not rare to find, one to one functions are

abundant in nature and they are reversible as well.

With this insight we will go to our next topic which is exponential functions.