Lec 46 - Exponential Functions: Graphing

Summary

This educational video script explores the properties of exponential functions, particularly focusing on functions of the form f(x) = a^x. It discusses the domain, range, intercepts, and end behavior of these functions, highlighting that for a > 1, the function is always increasing with no x-intercept, and for 0 < a < 1, it is decreasing. The script also emphasizes the importance of understanding the graphical behavior of exponential functions without relying solely on visual aids, and concludes with a summary table differentiating the characteristics of exponential functions based on the value of 'a'.

Takeaways

• 📚 The script discusses the properties of exponential functions, specifically focusing on functions of the form f(x) = a^x.
• 🔍 When 'a' is greater than 1, the domain of the function is the entire real line, and the range is from 0 to infinity.
• 📈 For a > 1, the function is always increasing and never touches the x-axis, meaning there is no x-intercept.
• 📉 When 0 < a < 1, the function is decreasing, and the end behavior is that as x approaches infinity, f(x) approaches 0, and as x approaches negative infinity, f(x) approaches infinity.
• 📍 The y-intercept for both cases (a > 1 and 0 < a < 1) is (0, 1), as a^0 equals 1 for any a.
• 🚫 There are no roots for exponential functions where a > 0, as the function never crosses or touches the x-axis.
• 💡 The horizontal asymptote for all such exponential functions is y = 0, indicating the behavior as x approaches infinity or negative infinity.
• 📊 The script introduces the concept of graphing exponential functions without the need for a graphing tool by understanding their properties.
• 🤓 The behavior of the function changes based on the value of 'a', and understanding this allows for predicting the shape and behavior of the graph.
• 🔑 The script uses specific examples, such as f(x) = 2^x and g(x) = 5^(-x), to illustrate the general properties of exponential functions.
• 📈📉 The reflection of the graph across the y-axis (by replacing x with -x) results in a change from an increasing to a decreasing function and vice versa.

Q & A

What is the domain of the function f(x) = 2^x?

The domain of the function f(x) = 2^x is the entire real line, as it is defined for all real values of x.

What is the range of the function f(x) = 2^x?

The range of the function f(x) = 2^x is from 0 to infinity (0 to ∞), since 2^x is always positive and never reaches 0.

Does the function f(x) = 2^x have any x-intercepts?

No, the function f(x) = 2^x does not have any x-intercepts because it never touches the x-axis, as it is always greater than 0.

What is the y-intercept of the function f(x) = 2^x?

The y-intercept of the function f(x) = 2^x is at the point (0, 1), which occurs when x = 0.

What is the end behavior of the function f(x) = 2^x as x approaches infinity?

As x approaches infinity, the function f(x) = 2^x also tends to infinity because the exponential growth rate is faster than linear growth.

What is the end behavior of the function f(x) = 2^x as x approaches negative infinity?

As x approaches negative infinity, the function f(x) = 2^x approaches 0, but never actually reaches it, due to the horizontal asymptote at y = 0.

Are there any roots for the function f(x) = 2^x?

No, there are no roots for the function f(x) = 2^x because it never crosses or touches the x-axis.

Is the function f(x) = 2^x increasing or decreasing throughout its domain?

The function f(x) = 2^x is increasing throughout its domain because for any x1 < x2, 2^x1 < 2^x2.

What happens when the base 'a' of the exponential function f(x) = a^x is greater than 1?

When the base 'a' is greater than 1, the function f(x) = a^x has similar properties to f(x) = 2^x: it is increasing, has no x-intercepts, and has a horizontal asymptote at y = 0.

What is the effect on the graph of the function when x is replaced with -x in an exponential function?

Replacing x with -x in an exponential function reflects the graph across the y-axis, changing the increasing function to a decreasing one and vice versa.

What is the general behavior of the graph of f(x) = a^x when 0 < a < 1?

When 0 < a < 1, the graph of f(x) = a^x is a decreasing function with the same domain and range as when a > 1, but with a horizontal asymptote at y = 0 and tending to infinity as x approaches negative infinity.

Outlines

00:00

📚 Introduction to Exponential Functions

This paragraph introduces the topic of exponential functions, emphasizing the importance of understanding their properties before analyzing specific examples. The domain of the function f(x) = 2^x is discussed, highlighting that it covers the entire real line. The range is identified as starting from 0 to infinity, with the function never reaching negative values. The concept of the y-intercept at (0, 1) is introduced, and the behavior of the function as x approaches positive and negative infinity is explored, noting that it does not touch the x-axis and has a horizontal asymptote at y=0.

05:07

📈 Analyzing the Graph and Behavior of Exponential Functions

The paragraph delves into the graphical representation and behavior of the exponential function f(x) = 2^x. It discusses the function's end behavior, noting that as x approaches infinity, the function also tends towards infinity, and as x approaches negative infinity, the function approaches 0. The absence of roots is established, and the function's monotonic increase across the entire real line is emphasized. Key points on the graph, such as the y-intercept and the point (1, 2), are identified, and the overall increasing nature of the function is reiterated.

10:09

🔍 Generalizing Exponential Functions with Bases Greater Than 1

This section generalizes the properties of exponential functions where the base 'a' is greater than 1. It asserts that all such functions share the same characteristics as 2^x, including the domain being the entire real line, the range from 0 to infinity, the absence of x-intercepts, and the horizontal asymptote at y=0. The paragraph also explains that the function's values change with different bases but the overall shape and behavior remain consistent, simplifying the process of graphing these functions.

15:15

📉 Reflecting on Exponential Functions with Bases Between 0 and 1

The final paragraph examines the behavior of exponential functions where the base 'a' is between 0 and 1, using 1/5^x as an example. It discusses the reflection of the graph across the y-axis, resulting in a decreasing function. The domain remains the real line, and the range is still from 0 to infinity. The y-intercept remains at (0, 1), but the end behavior is reversed, with the function approaching 0 as x approaches infinity and tending towards infinity as x approaches negative infinity. The paragraph concludes by summarizing the properties of these functions and introducing the concept of natural exponential functions for the next video.

Keywords

💡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' is greater than 0 and not equal to 1. In the video, the focus is on functions of the form 2^x and 1/5^x, which are used to illustrate the general behavior and properties of exponential functions. The script discusses the domain, range, and end behavior of these functions, which are central to understanding their graphical and algebraic characteristics.

💡Domain

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. In the context of the video, the domain of the exponential functions discussed is the entire set of real numbers (R), which means that you can input any real number into the function and get a valid output.

💡Range

The range of a function is the set of all possible output values (y-values) that result from the input values in the domain. For the exponential functions in the video, the range is (0, ∞), indicating that the output values are greater than 0 but never actually reach infinity, which is a key characteristic of exponential growth.

💡Y-Intercept

The y-intercept is the point where the graph of a function crosses the y-axis. In the video, it is mentioned that for the function f(x) = a^x, the y-intercept is always (0, 1), because any number raised to the power of 0 is 1. This is a specific example of how the script uses the concept of y-intercept to describe the behavior of exponential functions.

💡Horizontal Asymptote

A horizontal asymptote is a horizontal line that a function approaches but never touches as the independent variable (x) approaches infinity or negative infinity. In the video, it is explained that for exponential functions with a base greater than 1, y = 0 is a horizontal asymptote, meaning the function gets closer and closer to y = 0 as x goes to negative infinity but never actually reaches it.

💡End Behavior

End behavior refers to the behavior of a function as the input values approach the 'ends' of its domain, specifically as x approaches positive or negative infinity. The script describes the end behavior of exponential functions, noting that if the base is greater than 1, the function approaches infinity as x approaches positive infinity, and approaches 0 as x approaches negative infinity.

💡Roots

Roots of a function are the values of x for which the function equals zero. The video explains that exponential functions with a base greater than 0 and not equal to 1 do not have any roots, as they never touch or cross the x-axis, which is a fundamental property of these types of functions.

💡Increasing Function

An increasing function is one where the function's value increases as the input value increases. The video script explains that for exponential functions with a base greater than 1, the function is always increasing, meaning that as x gets larger, so does the value of the function.

💡Decreasing Function

A decreasing function is the opposite of an increasing function, where the function's value decreases as the input value increases. In the video, it is noted that when the base of the exponential function is between 0 and 1, the function is decreasing, which means that as x increases, the value of the function gets smaller.

💡Reflection Across Y-Axis

In the context of the video, reflection across the y-axis refers to the graphical transformation of a function when x is replaced with -x. This changes the direction of the function's end behavior, making an increasing function decreasing and vice versa. The script uses this concept to explain the behavior of the function 1/5^x, which is a reflection of 5^x across the y-axis.

💡Natural Exponential Function

The natural exponential function is a specific type of exponential function where the base is the mathematical constant e, which is approximately equal to 2.71828. The video script mentions that this topic will be introduced in a subsequent video, indicating its importance and relevance to the study of exponential functions.

Highlights

Introduction to the concept of exponential functions and the importance of understanding their properties.

Explanation of the domain of an exponential function, emphasizing the entire real line.

Analysis of the range of the function 2^x, highlighting it is always greater than 0 and approaches infinity.

Identification of the y-intercept at (0,1) for the function 2^x and its significance.

Discussion on the horizontal asymptote of the function 2^x, which is y=0.

Clarification that the function 2^x never touches the x-axis, indicating no x-intercept.

End behavior analysis of the function 2^x as x approaches both positive and negative infinity.

Investigation of the roots of the function 2^x, concluding there are none due to the function never touching zero.

Explanation of the function's monotonicity, asserting that 2^x is strictly increasing.

Graphical representation and identification of special points for the function 2^x.

Generalization of the properties for exponential functions where the base a is greater than 1.

Introduction to the reflection property of exponential functions when the base a is between 0 and 1.

Analysis of the function 1/5^x, discussing its reflection across the y-axis and its end behavior.

Identification of the y-intercept and the behavior of the function 1/5^x at x=1.

Summary of the properties of exponential functions based on the value of the base a.

Conclusion and transition to the topic of natural exponential functions in the next video.

Transcripts

Welcome back. So, I hope you must have done your exercises and you must have developed

some understanding about the exponential functions. Let us try to collect recollect that understanding

through 2 examples given here. So, let us first take 1 a which is f of x

is equal to 2 raised to x. If you have used DESMOS, you must have got the figure of the

function. But prior to receiving the figure of the function, let us see what should be

the domain of a function. We have already discussed in greater detail that the domain

of a function can be a real line, entire real line ok.

Now, if you look at this function which is 2 raised to x, this 2 is greater than 1 and

the 2 raised to x will always be greater than 2 raised to 0 which is equal to 1, 2 raised

to x is always greater than 2 raised to 0 whenever x is positive correct.

Now, because I x greater than 0, then 2 raised to x is always greater than 2 raised ok. So,

if x is less than 0, what will happen? 2 raised to x, when x is less than 0 will always be

less than 1. This is also possible. hm But when this 2 raised to; can this 2 raised to

x become negative? No. So, it is always greater than 0. So, if

you have this understanding, then you can easily write the function has a range which

is 0 to infinity. So, there is a split from when you consider a point 1, there is something

happening at point 0 comma 1 right. What is 0 comma 1? 0 comma 1 actually is an y intercept

ok, something is happening at 0 comma 1 because I have put 0 here for then it is I am getting

1. So, 0 comma 1 is also y intercept and there

is something happening which is going below 0. Is going below 1, your graph is going below

1 and therefore, this particular thing is going down, but it never goes below 0. This

is an interesting fact because if you consider 2 raised to x, it never goes below 0. It cannot

go to a negative number. Therefore, will it touch the x axis? It will

not touch x axis. In fact,x intercept is nil ok, but it is approaching 0. So, the something

that is approaching 0, so x intercept is actually it will never touch it; but it will actually

go along that line. So, this y is equal to 0, it will touch at infinity ok. So, such

a thing, we call as horizontal asymptote ok. oh

So, such a thing you call as horizontal asymptote. So, with this understanding, these these are

the things that I can make out directly without looking at the graph. So, let us now look

at the graph ok,before going to that, let us see what happens to the end behavior.

End behavior of a function as x tending to infinity. So, as 2 raised to x,you consider

a function 2 raised to x as x increases, this also increases. In fact it increases at a

rapid rate than x . So, this also should tend to infinity ok and as x tends to minus infinity,

we have already figured out y is equal to 0 is the horizontal asymptote. So, 2 raised

to x will actually go to 0 ok. Then, the question that we used to quantify

while consideringthe function, what are the roots of this function. So, do they have any

roots? In fact, using graphical method, it is very clear that it never touches 0. So,

there are no roots ok and the functionsincrease and decrease.

So, the domains of increase and decrease like polynomials, we studied domains of increase

and decrease; but here, I think my claim is no need to identify the domains of increase

and decrease. Why? Because you look at a function 2 raised to x, let us take x 1 not equal to

x 2 or x 1 lesser than x 2, without loss of generality, we can take this. Then, what what

can you say about 2 raised to x 1 and 2 raised to x 2?

See x 1 is less than x 2, so naturally ifit is raised to the power 2 power; 2 is raised

to power of x 1 and 2 is raised to power of x 2, this relation should hold. So, what I

am saying is the function is actually an increasing function and increasing functions are 1 to

1 . Therefore, I do not have any doubt that the increase and decrease, it is only increasing;

throughout the real line, the function is only increasing.

So, let us look at the graph of a function f of x is equal to 2 raised to x. Let us identify

the points. So, here you can identify a point right. So, this point we have seen as y intercept

and that point was 0 comma 1 right. Then, the one in this case, let us look at this

point which is 1 and where will it go? It will actually tell you 2. So,the point is

1 comma 2, the second point ok. So, these 2 points are very special points,

they tell you something. So, in particular, had it not been 2 raised to x, but a raise

to x, then that point would have been 1 comma a and if you mimic this graph over here y

x graph of y x is over here ok, this is a point 1, this is a point 0 and this is a point

which is a. So, that says a greater than 1; a greater

than 1, this relation is there, is greater than 0 yeah and therefore, the graph was a

point which lies here, which is here right. As x tends to infinity, this graph actually

goes to infinity; as x tends to minus infinity, this graph goes to 0.

These two points is these two point and this is an increasing function. As afrom left as

you come from left to right, it increases. So, this is an increasing function, y is equal

to 0 is the horizontal asymptote, that is very clear ok. The range of a function is

0 to infinity, that is also very clear. The domain of a function is entire real line .

So, we have got all the details necessary for finding this . Now, what it so special

about 2 raised to x, if I replace this 2 with 3, still I will have y intercept to be 0,

1 because 3 raised to 0 is also 1 and I will again have domain of f to be equal to R; range

of f to be equal to 0 infinity; no x intercept; y is equal to 0 will be horizontal asymptote;

x tends to infinity a raised to x tends to infinity, x tends to minus infinity, a raised

to x tends to 0. There are no roots. The function is only increasing.

And therefore, I will state this as a fact that every f of x is equal to a raised to

x, for a greater than 1 will have same properties as 2 raised to x. So, I do not there is no

need to draw different different values. The behavior is same only the values will

change. For example, in this case, where you have seen the graph of this 1, 2 is a point;

1, 2 is a point, suppose I consider 3 raised to x, 1 3 will be the point. So, only the

values are changing; but the shape, the behavior, everything else that is listed here remains

the same. Therefore, you do not have to draw a graph every time, only thing is you need

to evaluate the values in general. So, what is the graph of f of x equal to a

raised to x in general? It is this way for a greater than 1. So, remember that line that

we have drawn which is that the line for a, where we have eliminated these 2 points such

as 0, this is 1, we have identified what is the case for a greater than 1. You have also

identified the case, where a is less than 1 and greater than 0 . So, let us go back

and see what happens when 0 is less than a less than 1. So, if a lies here how is the

behavior? So, you have already analyzed. And let us take this function as g of x andtake

it to be g of x is equal to a raised to x and this is 1 by 5 raised to x. hm Now, you

do not really have to draw this graph, what you can do is ok. So, g of x is equal to 5

raised to minus x. So, here x is replaced by minus x.

So, what will be the change in the behavior? So, when x is replaced by minus x, you know

its reflection across y axis, you have solved many examples in the assignments. This y axis,

this is x; then when I put it as minus x, it will be simply reflected along y axis.

So, if you look at this graph and try to draw a graph of this function, then it should be

something like coming from here going here, it should be something like this, it should

actually look like a reflection along y axis. So, let us try to show it as reflection ok.

This will actually go very close, but never touch. So, let me erase this ok. So, this

is how it will look like. hm So, without actually thinking about anything else, you can simply

draw a graph of 1 by 5 raised to x; but still let us try to do it inregular set up.

So, what will be the domain of this function? The domain of this function is very clear

becausewe have used it several times, the domain of this function will be real line.

Range, nothing changes; 0, infinity because it is a reflection across y axis. So, let

us look at this function. So, the domain will be R; range will be 0 to infinity. What will

be the y intercept? Because it is a reflection, so y intercept would not change, so it will

be 0, 1 only. x intercept will be nil, there would not be any x intercept.

And therefore, no roots and what is what about the end behavior? End behavior is like x tending

to infinity, x tending to minus infinity . So, when x tends to infinity, the end behavior

will be because it is a reflection you see. So, when x was tending toinfinity there, it

was going to infinity. So, and x tending to minus infinity, function

5 raised to x would have behaved, it will go to 0. So, that reflection will make this

a raised to x or 5, 1 by 5 raised to x whatever is the function 1 by5 raised to x, let me

do it properly. So, this will make 1 by 5 raised to x to go

to 0 and this function 1 by 5 raised to x will go to infinity ok. Good. Then, because

it is a reflection, the increasing thing will become decreasing. So, there is nointelligence

here. So, this will be in fact a decreasing function wonderful. So, we have analyzed everything

without taking much efforts. This is the beauty of once you understand the functions on graphical

plane. So, here is the graph of a function which

is given to us 1 by 5 raised to x, you also might have plotted and naturally, the we will

analyze whether it coincides with our thing. So, this is a point 0 comma 1, now it is 1

by 5. So, your point will be somewhere here, sorrythis is 5. So, the point 1 is here and

this point is 1 by 5. So, 1 comma 1 by 5, this is done. Then, as

x tends to infinity, as x tends to infinity, this function goes to 0. As x tends to minus

infinity that is this way, this function actually goes to infinity ok and this function is decreasing.

From left to right if you come, you are actually coming down. So, it is a decreasing function.

So, this completely gives us an understanding of what the graph of a function will look

like. Also, the same fact is true that every f x

a raised to x, where 0 is less than a is less than 1 has same properties as 1 by 5 raised

to x. Therefore, it is a representative class. So, you do not have to worry about the because

it is a representative class, you have to worry about all other functions. All other

functions will have a similar behavior. So, we have done a lot, let us summarize these

things in a neat table which is this. So, this is the summary ofthe table . So, if I

have been given a function f of x is equal to a raised to x, thento be more precise,

let me draw a line here. This is a line; it does not look like a line, but assume that

this is a line. This is the point 1, then I am talking about 0 less than a less than

1 that this zone. In this zone, the domain of a function is

R; range of a function is 0 to infinity. There are no x intercepts, no; y intercept is 0,

1. Horizontal asymptote y is equal to 0 is there. The function is decreasing. The end

behavior as x tends to infinity, f of x tends to 0; as x tends to minus infinity, f of x

tends to infinity correct. Then,you look at the function which is a greater

than 1, domain is real line, range is 0 infinity, nil; 0, 1, y intercept is 0, 1. Horizontal

asymptote is y is equal to 0. The only distinguishing feature is the function is increasing here

and a function is decreasing here and because it is increasing and decreasing, the end behavior

changes that is because it is decreasing, it will decrease to 0 because it is bounded

below by 0 and because this is increasing, it will increase to infinity, but here it

will go to 0 ok. Then finally, you see the prototypes, just

look at the graphs of these two functions ok. This ends our topic on exponential functions.

Now, we will introduce something which is called natural exponential function in the

next video.