Rational Function (Domain, x & y - Intercepts, Zeros, Vertical and Horizontal Asymptotes and Hole)

Summary

This tutorial video teaches viewers how to analyze rational functions by determining their domain, x-intercepts, y-intercept, zeros, vertical and horizontal asymptotes, and holes. The instructor uses step-by-step examples to illustrate the process of finding these characteristics, starting with defining the domain as all real numbers except the zeros of the denominator. The video then demonstrates how to find x and y-intercepts by setting y to zero and x to zero respectively, and explains how to identify zeros and asymptotes based on the degrees of the numerator and denominator. The concept of holes, or removable discontinuities, is also covered, showing how they occur when there's a common factor in the numerator and denominator. The tutorial is designed to provide a comprehensive understanding of rational functions.

Takeaways

• 📘 A rational function is defined as the ratio of two polynomial functions.
• 🚫 The domain of a function includes all real numbers except the zeros of the denominator.
• ✖️ To find the x-intercept, set y to zero and solve for x, ensuring x is within the domain.
• 🔍 The y-intercept is found by setting x to zero and solving for y.
• 🔢 The zeros of a rational function are the values of x that make the function equal to zero, considering the domain restrictions.
• 📉 Vertical asymptotes occur at values of x that make the denominator zero, but are not included in the domain.
• 📈 The horizontal asymptote is determined by comparing the degrees of the numerator and the denominator, and can be none if the degree of the numerator is greater.
• 🕳 A hole, or removable discontinuity, is present when there is a common factor in the numerator and denominator, leading to a discontinuity at the value of x that makes the common factor zero.
• 🔄 The process involves factoring the numerator and denominator to identify intercepts, asymptotes, zeros, and holes.
• 📚 The tutorial provides a systematic approach to analyzing rational functions, emphasizing the importance of domain considerations.

Q & A

What is a rational function?

A rational function is any function that can be written as the ratio of two polynomial functions, with the numerator and the denominator being polynomials.

How do you find the domain of a rational function?

The domain of a rational function is the set of all real numbers except for the values that make the denominator zero.

What is an x-intercept and how do you find it?

An x-intercept is a point where the graph of the function crosses the x-axis, which means the y-value is zero. To find it, set y to zero and solve the equation for x.

What is a y-intercept and how is it determined?

A y-intercept is the point where the graph of the function crosses the y-axis, which occurs when x is zero. Determine it by substituting x with zero in the function and solving for y.

What are the zeros of a rational function?

The zeros of a rational function are the values of x that make the function equal to zero. They are typically found by setting the function to zero and solving for x.

How do you determine if a rational function has a vertical asymptote?

A rational function has a vertical asymptote at values of x that make the denominator zero but are not included in the domain of the function.

What is a horizontal asymptote and how is it found?

A horizontal asymptote is a horizontal line that the graph of the function approaches as x approaches infinity or negative infinity. It is found by comparing the degrees of the numerator and the denominator.

What is meant by a 'hole' in a rational function?

A 'hole' in a rational function refers to a removable discontinuity, which is a point where the function is not defined due to a common factor in the numerator and denominator that can be canceled out.

How do you find the coordinates of a 'hole' in a rational function?

To find the coordinates of a 'hole', identify the common factor in the numerator and denominator, set it equal to zero to find the x-value, and then substitute this x-value back into the function to find the y-value.

What is the significance of the degrees of the numerator and denominator in determining the horizontal asymptote?

The degrees of the numerator and denominator determine the horizontal asymptote as follows: if the degree of the numerator is less than the degree of the denominator, the asymptote is y=0; if they are equal, the asymptote is the ratio of the leading coefficients; if the degree of the numerator is greater, there is no horizontal asymptote.

Outlines

00:00

📘 Introduction to Rational Functions

This paragraph introduces the concept of rational functions, defined as the ratio of two polynomial functions. The tutorial aims to teach viewers how to find various aspects of a rational function, including its domain, x-intercepts, y-intercepts, zeros, vertical and horizontal asymptotes, and holes. The first example provided is a function f(x) = (x^2 - x - 6) / (x - 3), and the process begins with determining the domain by setting the denominator equal to zero and solving for x, which results in x = 3. The domain is all real numbers except x ≠ 3.

05:00

📐 Finding X-Intercepts and Y-Intercepts

The tutorial continues with the process of finding the x-intercepts by setting y to zero and solving the equation. The function is factored to simplify the process, resulting in (x - 3)(x + 2) / (x - 3). It's noted that x = 3 is not included in the domain, so the x-intercept is at x = -2, y = 0. The y-intercept is found by setting x to zero, leading to y = 2. The zeros of the function are the same as the x-intercepts, but since x = 3 is not in the domain, it's not considered a zero. The concept of horizontal asymptotes is introduced, and it's determined that there is no horizontal asymptote for this function because the degree of the numerator is greater than the degree of the denominator.

10:02

🔍 Analyzing Rational Functions for Asymptotes and Holes

The tutorial moves on to discuss vertical asymptotes, which are the values of x that are not in the domain. For the given function, x = 3 is a vertical asymptote. The concept of holes is introduced, which are removable discontinuities caused by common factors in the numerator and denominator. Since there is a common factor of (x - 3), x = 3 is identified as a hole, and by substituting x = 3 into the remaining factor (x + 2), the y-coordinate of the hole is found to be 5, making the hole's coordinates (3, 5).

15:04

📉 Exploring More Rational Functions and Their Characteristics

A second example is presented, f(x) = 1 / (x - 1), to illustrate finding the domain, x-intercepts, y-intercepts, and asymptotes without common factors. The domain is all real numbers except x ≠ 1. There is no x-intercept because the equation 0 = 1 / (x - 1) has no solution for x. The y-intercept is found by setting x to zero, resulting in y = -1. Since there are no common factors, there are no zeros or holes. The vertical asymptote is at x = 1, and the horizontal asymptote is y = 0 because the degree of the numerator is less than the degree of the denominator.

20:08

📚 Comprehensive Review of Rational Functions

The final example, f(x) = (x^2 - 4) / (x^2 - 9), is used to demonstrate finding the domain, x-intercepts, y-intercepts, zeros, vertical asymptotes, and horizontal asymptotes. The domain excludes x = ±3. The x-intercepts are found to be x = -2 and x = 2. The y-intercept is calculated as y = 4/9. The zeros are the same as the x-intercepts. Vertical asymptotes are at x = ±3, and the horizontal asymptote is y = 1, as the degrees of the numerator and denominator are equal. The tutorial concludes with a summary of the methods to analyze rational functions.

Keywords

💡Rational Function

A rational function is a mathematical expression that can be written as the ratio of two polynomial functions. In the context of the video, the theme revolves around understanding and analyzing rational functions by examining their properties such as domain, intercepts, and asymptotes. The script provides an example of a rational function f(x) = (x^2 - x - 6) / (x - 3), where the numerator is a quadratic polynomial and the denominator is a linear polynomial.

💡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 video, the domain is determined by identifying the values that make the denominator of a rational function equal to zero, as these values would make the function undefined. For instance, in the example f(x) = (x^2 - x - 6) / (x - 3), the domain is all real numbers except x = 3, since this is the value that would make the denominator zero.

💡X-intercept

The x-intercept is the point where the graph of a function crosses or touches the x-axis. In the script, finding the x-intercept involves setting the function equal to zero and solving for x. For the given rational function, the x-intercepts are calculated by setting y = 0 and solving the resulting equation, yielding the points where the graph intersects the x-axis.

💡Y-intercept

The y-intercept is the point where the graph of a function crosses or touches the y-axis. This is found by setting x to zero in the function and solving for y. In the video script, the y-intercept is determined by substituting x = 0 in the function f(x) and calculating the corresponding y-value, which is the point where the graph intersects the y-axis.

💡Zeros

Zeros of a function are the x-values for which the function's output is zero. In the context of the video, zeros are calculated similarly to x-intercepts, by setting the function equal to zero and solving for x. However, zeros are specifically relevant to rational functions where the numerator is set to zero, excluding any values that would make the function undefined due to the domain restrictions.

💡Vertical Asymptote

A vertical asymptote is a vertical line that the graph of a function approaches but never touches. It occurs where the denominator of a rational function is zero, but the numerator is not zero. In the video, vertical asymptotes are identified by the values of x that make the denominator zero, provided these values are within the domain of the function.

💡Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches positive or negative infinity. The script explains how to determine horizontal asymptotes based on the degrees of the numerator and denominator of the rational function. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degrees are equal, the asymptote is the ratio of the leading coefficients of the numerator and denominator.

💡Whole

A whole, also known as a removable discontinuity, is a point where the graph of a rational function appears to have a hole or discontinuity due to a common factor in the numerator and denominator. In the video, the concept of a whole is explained by identifying common factors in the numerator and denominator and then solving for the x-value that would make the function undefined if not for the common factor, which can be canceled out to 'remove' the hole.

💡Factoring

Factoring is the process of breaking down a polynomial into a product of its factors. In the video, factoring is used as a method to simplify the rational function and to find the x-intercepts and zeros by setting the factored form of the numerator equal to zero. The script demonstrates factoring in the context of finding the x-intercepts for the function f(x) = (x^2 - x - 6) / (x - 3).

💡Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. In the video, the degree of the polynomial in the numerator and denominator is used to determine the presence and position of horizontal asymptotes. The script explains that if the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote, which is a key concept in analyzing the end behavior of rational functions.

Highlights

Definition of a rational function as the ratio of two polynomial functions.

Explanation of finding the domain of a rational function by excluding the zeros of the denominator.

Method to determine the x-intercept by setting y to zero and solving the equation.

Procedure for finding the y-intercept by substituting x with zero in the function.

Technique to identify the zeros of a rational function, which are the same as the x-intercepts.

Criterion for determining the presence of a horizontal asymptote based on the degrees of the numerator and denominator.

Identification of vertical asymptotes as the values that make the denominator zero, excluding them from the domain.

Concept of a hole or removable discontinuity in a rational function where a common factor exists in both the numerator and denominator.

Tutorial on solving for the domain, x-intercept, y-intercept, zeros, vertical, and horizontal asymptotes, and holes for a given rational function.

Example problem demonstrating the calculation of the y-intercept resulting in a value of 2.

Example illustrating that there is no x-intercept when the function results in an undefined value.

Explanation of how to find the vertical asymptote by identifying the values that are excluded from the domain.

Guide on calculating the horizontal asymptote by comparing the degrees of the numerator and the denominator.

Process for determining the hole in a rational function by factoring and identifying common factors.

Second example problem walkthrough for a different rational function, emphasizing the steps to find various characteristics.

Final summary of the tutorial's content, reinforcing the understanding of rational functions and their key features.

Transcripts

in this tutorial video i will be

teaching you how to get

the domain

the x-intercepts

the y-intercepts the zeros

the vertical asymptote the horizontal

asymptote and the whole of rational

functions

let us first define what is a rational

function

a rational function is any function that

can be written as the ratio

so when we say ratio we have the

numerator and denominator of two

polynomial functions

now let's have our problem number one

let's say we have f of x is equal to x

squared minus x minus six all over x

minus three

we're going to find the domain the

x-intercept y-intercept zeros the

vertical asymptote horizontal asymptote

and the whole if you notice we have

guide in every item

so for the domain

so let's find first the domain

the domain is the set of all numbers

except the zeros of the denominator so

for the domain

let's say a

domain

find the zeros of our denominator

so our denominator here is

x minus three

so copy x minus three just equate to

zero

so solve for x so we have

x is equal to

opacity of three

so our domain is

x such that x is an element

of real numbers

exact

exactly

exact

3

or simply just get the value for x to

make our rational function undefined so

get our denominator then equate to 0.

letter b

x intercept

our guide to find the x intercept let y

be equal to zero

so f of x that stands for our y

so we have

y

f of x is our y

then we have

x squared minus

x minus six

all over

x minus three

let y be equal to zero so

y be zero

and we have now

x squared minus x minus six

all over

x minus three

cross multiply so we have zero times x

minus three that will become zero is

equal to

x squared minus x minus six

now let us factor x squared minus x

minus six

okay to get the factor

um get the product of negative negative

six the sum must be negative one

so factoring general trinomial so x and

x

factors of negative six

we have negative three and positive two

let's check

negative three times positive two that

is negative six

negative three plus two that is negative

one or simply negative x here

so

this function

is just equal to x minus three times x

plus two all over x minus three

okay let us write here go back in letter

b zero is equal to

just copy the factored form

x minus 3 and x plus 2.

now

equate to 0

so we have x minus 3 is equal to 0

and

x plus two is equal to zero

solve for x

so x is equal to

positive three

and

here

x is equal to negative two

so we have two x-intercepts we have

three zero

and

negative two

zero

but take a look in our

domain that our x must not be equal

to three

so

we're just going to consider negative

two

zero

so our x-intercept is negative two

zero

where did we get zero

that is the value of our y

so

our uh let's have a short review

our x-intercept is a point that means it

must contain our

[Music]

abscissa and ordinate so our x here is

negative two then y

zero

now letter c

y-intercept

y-intercept

for the y-intercept

let x be equal to zero

so we have

let x be equal to zero

so

y

is equal to

just substitute

our x to zero so we have

zero squared minus zero

change x to zero

minus six all over

zero minus three

now we have

zero square that is zero minus zero

that is zero minus six so negative six

all over zero minus three negative three

negative six divided by negative three

that is positive 2.

so our y intercept is

our x is 0 so we have 0

positive 2.

this will be our x intercept

okay or y is equal to two

next

letter d

now let's have the zeros of our rational

fraction

zeros

in our guide this is the same as our

x-intercept so letter b x-intercept we

have our solution here

and we're going to find the value for

our x

so x here is equal to negative 2 and

positive 3 but take a look but take a

look in our domain

our restricted value for x is three so

x is a restricted value for our zeros

x is equal to

negative

discuss here because this is a crucial

part

let's have

our first

condition

d here stands for the degree

if the degree of the numerator and

stands for the numerator if the degree

of the numerator is less than the degree

of the denominator

our y is equal to zero

if the degree

of the numerator is equal to the degree

of the denominator we're going to use

a over b

where a is our leading coefficient in

the numerator and b is the leading

coefficient in that denominator

and if the degree of the numerator is

greater than the degree of the

denominator we don't have a horizontal

asymptote

now studying our given

the degree

of our numerator

is greater than

the denominator

in our numerator earlier our quadratic

so our degree in the numerator is 2

and in our denominator that is a linear

function

so 1

so

the degree of the numerator is greater

than the degree of our denominator

so horizontal asymptote in this case we

don't have

none

there is no horizontal asymptote

now let's have

g

the whole

okay

hole is also known as the removable

discontinuities

so these are the values or the inputs

that causes our numerator and

denominator b zero

so in this case notice that

we have a common factor

x minus 3

x minus 3 in our numerator and in the

denominator so

these are the discontinuity of our

function so get the common factor

for the whole we have x minus 3 is equal

to 0

that is the common factor

so find the value for x so x is equal to

3

now we have

our abscissa or our x

to find our y

get the remaining factor in our

numerator which is x plus two

so x plus two

now

substitute the value of our x which is

three so we have three plus two

now we have three plus two we have five

now the coordinate of our whole

is

x is 3

then this will be the value of our y

[Music]

3 comma 5

that would be the discontinuities of our

rational function 3 comma 5 this is the

whole

again

if you have a common factor in the

numerator and denominator

get the common factor then equate to 0

then solve for x and after that

substitute to get our

organic

okay that is on how to find the domain

x-intercept y-intercept zeros vertical

asymptotes horizontal asymptote and

whole let's have our second example

now let's have our second example

f of x is equal to one over x minus one

let us find the domain

so letter a

domain

so get the denominator x minus one

equate to zero so x is equal to one

so our domain is

x such that x is an element of real

numbers

except

positive one

b

the x-intercept for the x-intercept let

y be equal to zero so x intercept

so

y

is equal to

one over x minus one again f of x is the

same as our y so

let y be equal to zero so we have zero

is equal to

one over

x minus one

cross multiply

so we have zero is equal to one notice

that we don't have a value for x

because zero times x minus one is zero

so that means we don't have

no

x

intercept

or simply none for the x-intercept

letter c

y-intercept

for the y intercept let x be equal to

zero

so we have

y is equal to one minus

x minus one

let x be zero so we have y is equal to

one over

zero minus one

that is

one over negative one

therefore y is equal to negative one

so our y intercept is

zero comma

negative one

that is our y y-intercept

and for the zeros

letter d

zeros

the same as the x-intercept so we don't

have

x-intercept so 0's

9

and for the vertical asymptote

letter e

these are the restricted values for x

vertical asymptotes

according to our domain

x such that x is an element of real

numbers except one

so vertical asymptote

x is equal to

one

letter f

horizontal asymptote

1 over x minus 1. so study our condition

if the degree of the numerator is less

than the degree of the denominator

our horizontal asymptote is y is equal

to zero

so

we have here a degree of one

in our denominator that means

the degree of the

denominator is greater than the degree

of the numerator

so we have our first condition

y is equal to zero

and lastly

to find the whole

we don't have a common factor

both in numerator and denominator

so

f of x is equal to x squared minus four

all over x squared minus nine

okay let us find first the factored form

of our given

in our numerator

that is

x plus 2

x minus 2

and in our denominator

we have

x plus three

and x minus three

different of two squares

now get the domain

domain

our denominator

is x squared minus nine that is

the same as quantity x plus three

times

x minus three so that is

x plus three

and x minus three equate to zero

now solve for x

we have x plus three is equal to zero

and x minus three is equal to zero

so we have x is equal to negative 3

and x is equal to positive 3.

for our domain

x such that x is an element of real

numbers

except

positive negative 2

and for the x-intercept letter b

let y be equal to zero

so

f of x that stands for our y so

zero

is equal to

x squared minus four all over

x squared minus nine

cross multiply

we have zero is equal to

x squared minus four

get the factored form of x squared minus

nine so that is zero is equal to

x squared minus four

factors are

x plus two

x minus two

now equate to zero

so

x plus two is equal to zero

the other one x minus two is equal to

zero

so our x-intercept we have two values we

have

negative 2 and

positive 2.

okay

that is our x-intercept

and

letter c y-intercept

okay

letter c

y intercept let x be equal to zero

so in this case

y is equal to

let x be zero so zero squared minus four

all over

zero squared minus nine

so that is negative four all over

negative nine

negative divided by negative

that is

four over nine

so y is equal to four over nine

so

we can write this as 0

4 over 9 so that you can easily plot the

points and for this one we have

x is negative 2 0

and

two zero

okay

now get the zeros letter d

zeros

for the zeros

same as the x intercept so letter b x

intercept

so our zeros x is equal to

positive negative two

next letter e

vertical asymptote

okay

for the vertical asymptote

according to our domain our restricted

values

are positive and negative three

so for the vertical asymptote we have x

is equal to

positive three

and x is equal to negative

[Applause]

those are

the vertical asymptotes

and letter f

horizontal

so examine our condition

condition one if the degree of the

numerator is less than the degree of the

debit the denominator

that is

y is equal to zero

so we study our given we have the same

degree

so our degree

of the numerator is equal to the degree

of the denominator so we're going to use

y is equal to

a over b

wherein

a is the leading coefficient in the

numerator in this case you have one

over

b is the leading coefficient in the

denominator

we have also one one divided by one

that is one

so our horizontal asymptote is y is

equal to

one and our whole

as you can see in our factor factored

form we don't have a common factor so we

don't have

i hope you understand the rational

function and how to get

the domain x intercept y intercept the

zeros vertical asymptotes horizontal and

the whole

thank you for watching senior pablo tv