How to find the domain and the range of a rational function?

Summary

This script explains the process of determining the domain and range of a function, focusing on a specific example. It emphasizes the importance of identifying restrictions, such as where the function is undefined, and how to factor and cancel terms to simplify the function. The script also discusses graphing the function, including the y-intercept and slope, and the significance of open circles on the graph to represent points where the function is not defined. Finally, it touches on the concept of limits to find missing y-values at restricted points.

Takeaways

• 🔍 The domain of the function is determined by setting the bottom of the fraction to zero, resulting in x ≠ 3.
• 📐 The domain is expressed as two intervals: (-∞, 3) U (3, ∞), excluding the point where x = 3.
• 🔢 Factoring and canceling terms in the function simplifies it to x + 1/4, indicating a linear relationship.
• ✏️ The function is not a straight line due to the restriction x ≠ 3, which must be considered in the graph.
• 📊 The graph of the function starts with the y-intercept at 1/4 and has a slope of 1/4, representing a line with a gentle incline.
• 🚫 At x = 3, the function is undefined, so an open circle is used on the graph to denote this exclusion.
• 📈 The y-value at the open circle (x = 3) is found by substituting x = 3 into the original function, resulting in y = 1.
• 📋 The range of the function is from negative infinity to just below 1, and then from 1 to infinity, with a hole at y = 1.
• 🔄 The concept of limits is introduced to understand the behavior of the function around the excluded value x = 3.
• 📝 The final graph of the function includes a hole at (3, 1), indicating that 3 is not in the domain and 1 is not in the range.

Q & A

What is the first step to determine the domain of the function described in the script?

The first step is to set the denominator of the function equal to zero and solve for x, ensuring that the denominator does not equal zero in the domain.

What is the restriction for the domain of the function based on the script?

The domain is restricted such that x cannot be equal to 3, as this would make the denominator zero.

How is the domain of the function described in the script?

The domain is described as all real numbers from negative infinity to 3, not including 3, and from 3 to infinity.

What is the process to find the range of the function as explained in the script?

The range is found by factoring and canceling terms in the numerator and denominator, resulting in a simplified function of (x + 1)/4.

Why is the function not considered a straight line according to the script?

The function is not a straight line because it has a discontinuity at x = 3, where the term x - 3 is canceled out, indicating that x cannot be equal to 3.

What is the y-intercept of the function as described in the script?

The y-intercept is 1/4, which is found by setting x to 0 in the simplified function.

What is the slope of the line when graphing the function from the script?

The slope of the line is 1/4, indicating that for every 1 unit increase in x, y increases by 1/4 unit.

How does the script suggest to handle the discontinuity at x = 3 when graphing the function?

The script suggests to mark x = 3 with an open circle on the graph and calculate the y-value at this point using the limit concept to understand the behavior of the function around 3.

What is the y-value at the point of discontinuity (x = 3) as per the script?

The y-value at x = 3 is 1, which is calculated by substituting x = 3 into the original function before simplification.

How is the range of the function described in the script?

The range of the function is all real numbers from negative infinity to 1, not including 1, and from 1 to infinity.

What is the significance of the open circle at (3, 1) on the graph as mentioned in the script?

The open circle at (3, 1) signifies that the point is not part of the function's graph because it corresponds to a discontinuity where the function is undefined.

Outlines

00:00

📐 Understanding Domain and Range of a Function

This paragraph discusses the process of determining the domain and range of a mathematical function. The focus is on a specific function where the domain is found by setting the denominator equal to zero and solving for x, resulting in restrictions at x=3. The range is determined by factoring and canceling terms, leading to a simplified expression of the function. The function is not a straight line due to the cancellation of terms, which introduces a hole at x=3. The graphing of the function includes finding the y-intercept and using the slope to sketch the line, with a special note to exclude the point where x=3, represented as an open circle on the graph.

Keywords

💡Domain

In the context of the video, 'domain' refers to the set of all possible input values (x-values) for which a function is defined. The video script discusses determining the domain by setting the denominator of a fraction to not equal zero, which is a fundamental step in calculus to avoid division by zero. For instance, the script mentions 'x cannot be equal to 3' as a restriction on the domain, indicating that the function is not defined at x = 3.

💡Range

The 'range' of a function is the set of all possible output values (y-values) that result from the function. The video explains how to find the range by factoring and canceling terms in a rational function, which simplifies the expression and helps in determining the possible y-values. The script uses the example of the function simplifying to 'X Plus 1 over 4' to illustrate how the range is derived.

💡Factoring

Factoring is a mathematical technique used to break down expressions into product forms. In the video, factoring is used to simplify the function and to find the domain and range. The script specifically mentions factoring the numerator and the denominator of the function, which allows for terms to be canceled out, a critical step in determining the simplified form of the function.

💡Graphing

Graphing is the visual representation of a function's behavior. The video script describes the process of graphing a function, starting with the y-intercept and using the slope to plot additional points. It emphasizes the importance of graphing to visualize the function's behavior, including the open circle at a point where the function is not defined, such as at x = 3.

💡Y-intercept

The 'y-intercept' is the point where a function crosses the y-axis. In the video, the y-intercept is calculated as 'one over four' and is used as a starting point for graphing the function. This is a basic concept in graphing linear equations and is crucial for plotting the initial point of the function on the coordinate plane.

💡Slope

Slope in the context of the video refers to the rate of change of the function's y-values with respect to x-values. The script mentions a slope of 'one over four,' indicating that for every one unit increase in x, the y-value increases by a quarter. This is essential for graphing as it dictates the direction and steepness of the line.

💡Open Circle

An 'open circle' on a graph represents a point that the function approaches but does not actually reach. The video script discusses the significance of the open circle at x = 3, where the function has a hole, indicating that the function is not defined at this x-value. This is a key concept in understanding the continuity and discontinuity of functions.

💡Limit

The 'limit' in calculus is a fundamental concept that describes the behavior of a function as the input approaches a certain value. The video script touches on the concept of limits when discussing the y-value at the point where the function has a hole (x = 3). It uses the limit to find the y-value that the function approaches but does not attain, which is crucial for understanding the function's behavior near points of discontinuity.

💡Continuity

Continuity in a function refers to the property where the function has no breaks, holes, or jumps in its graph. The video script discusses continuity in the context of the function having a hole at x = 3, which means it is discontinuous at that point. Understanding continuity is essential for analyzing the smoothness and完整性 of a function's graph.

💡Rational Function

A 'rational function' is a function that is the ratio of two polynomial functions. The video script deals with a rational function where the domain and range are determined by analyzing the polynomials in the numerator and the denominator. Rational functions are a common topic in calculus and algebra, and understanding their properties, such as where they are undefined, is crucial for their analysis.

Highlights

Identifying the domain by setting the bottom of the fraction to zero and solving for x.

Domain restriction where x cannot be equal to 3.

Domain is from negative infinity to 3, not including 3, and then from 3 to infinity.

Factoring and canceling out terms in the numerator and denominator to simplify the function.

Simplification results in a function of the form (x + 1) / 4.

The function is not a straight line due to the restriction x cannot be equal to 3.

Graphing the function as a line with a slope of 1/4 and y-intercept of 1/4.

Considering the domain restriction when graphing, marking an open circle at x = 3.

Determining the y-value at the point of discontinuity (x = 3) using the concept of limits.

Calculating the y-value at x = 3 to be 1, which is the missing y-value.

Defining the range of the function as extending from negative infinity to just below 1, and then from 1 to infinity.

Noting the open circle hole at (3, 1) on the graph, indicating x = 3 is not in the domain.

Identifying 1 as the maximum y-value, which is not in the range due to the discontinuity.

Emphasizing the importance of graphing to understand the domain and range of a function.

The function's behavior around the point of discontinuity is crucial for understanding its limits.

Transcripts

function and let me tell you this right here  can be really tricky so pay close attention  

first off for the domain remember you always  look at the bottom and then 39 equal to zero  

so we have the 4X minus 12. go ahead and just  put 4X minus 12 and we do not want this to be  

equal to zero and then work this out put a  12 here so we get 4X it's not equal to zero  

for x minus 4X is 9 equal to 12 and then d by  posits by 4 x cannot be equal to three and that's  

the Restriction so we'll just write the domain  is that we go from negative Infinity to 3 but not  

including the three but Union the other part is  three to Infinity this is it done deal now for the  

range this right here is usually harder but for  this one yeah it's also tricky too notice on the  

top and bottom we can actually factor and cancel  things out on the top we Factor this out we get  

x minus 3 times X plus one on the bottom we can  Factor all four and that will give us four minus  

4 times x minus three and this and that can cancel  so we are just going to get X Plus 1 over 4.

in fact this is just a line right  because we can look at this as 1 over 4X  

plus 1 over 4. however this function is  not really just a line because when we  

cancel the x minus 3 on the top and bottom we  have to denote that X cannot be equal to 3.  

when we graph this line in fact we have to put  this into consideration as well so let me show you  

okay so let's review how we can graph the line  we start with the Y intercept which is one over  

four that says right here all right and then  the slope is one over four that means we go up  

one time and then you go to a row four times  it doesn't really matter how the graph looks  

like because you know you will just have a  line like this just give a quick sketch and  

then of course you know the line will keep on  going down forever and keep going up forever  

now here's the deal X cannot be equal to 3 we  will also have to put that into consideration on  

the graph so let's say three is right here because  earlier it was up one to the right four times yep  

now we are going to go up and then carry off that  point on the graph so we will have an open circle  

here and here is the key we have to find out the  Y value of this open circle because that's the Y  

value that's missing and what can be anything else  except for that so how can we do it well in this  

case that's the concept of the limit we are going  to study the behavior of the function around 3.  

what we can do is actually just go ahead and put a  three not to the original though because otherwise  

we will get 0 over 0 [Music] . go ahead and just  put it here so if you have this right here one  

over four times three and then plus one over four  we can work this out this right here is just going  

to be three over four plus one over four and  that will just give us 4 over 4 which is one  

so in fact when X is equal to 3  the Y value here it will be 1.  

and that's actually the missing y value and when  you have the picture you know to find the domain  

so much better you know have a picture first right  here I can tell you the range will just go into B  

we go for negative Infinity right because  y can go all the way down up to one  

right once missing so stop like this and then  Union we go from one to Infinity all right

that's it and perhaps I will tell you guys this  right here uh there's a open circle hole at three  

comma one and that's why 3 is not in the domain  and likewise one is also nine the range that's it