How to find the domain and the range of a rational function?
Summary
TLDRThis 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
- š The domain of the function is determined by setting the bottom of the fraction to zero, resulting in x ā 3.
- š The domain is expressed as two intervals: (-ā, 3) U (3, ā), excluding the point where x = 3.
- š¢ Factoring and canceling terms in the function simplifies it to x + 1/4, indicating a linear relationship.
- āļø The function is not a straight line due to the restriction x ā 3, which must be considered in the graph.
- š 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.
- š« At x = 3, the function is undefined, so an open circle is used on the graph to denote this exclusion.
- š The y-value at the open circle (x = 3) is found by substituting x = 3 into the original function, resulting in y = 1.
- š 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.
- š The concept of limits is introduced to understand the behavior of the function around the excluded value x = 3.
- š 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
š 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.
Mindmap
Keywords
š”Domain
š”Range
š”Factoring
š”Graphing
š”Y-intercept
š”Slope
š”Open Circle
š”Limit
š”Continuity
š”Rational Function
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 hereĀ can be really tricky so pay close attentionĀ Ā
first off for the domain remember you alwaysĀ look at the bottom and then 39 equal to zeroĀ Ā
so we have the 4X minus 12. go ahead and justĀ put 4X minus 12 and we do not want this to beĀ Ā
equal to zero and then work this out put aĀ 12 here so we get 4X it's not equal to zeroĀ Ā
for x minus 4X is 9 equal to 12 and then d byĀ posits by 4 x cannot be equal to three and that'sĀ Ā
the Restriction so we'll just write the domainĀ is that we go from negative Infinity to 3 but notĀ Ā
including the three but Union the other part isĀ three to Infinity this is it done deal now for theĀ Ā
range this right here is usually harder but forĀ this one yeah it's also tricky too notice on theĀ Ā
top and bottom we can actually factor and cancelĀ things out on the top we Factor this out we getĀ Ā
x minus 3 times X plus one on the bottom we canĀ Factor all four and that will give us four minusĀ Ā
4 times x minus three and this and that can cancelĀ so we are just going to get X Plus 1 over 4.
in fact this is just a line rightĀ because we can look at this as 1 over 4XĀ Ā
plus 1 over 4. however this function isĀ not really just a line because when weĀ Ā
cancel the x minus 3 on the top and bottom weĀ have to denote that X cannot be equal to 3.Ā Ā
when we graph this line in fact we have to putĀ this into consideration as well so let me show youĀ Ā
okay so let's review how we can graph the lineĀ we start with the Y intercept which is one overĀ Ā
four that says right here all right and thenĀ the slope is one over four that means we go upĀ Ā
one time and then you go to a row four timesĀ it doesn't really matter how the graph looksĀ Ā
like because you know you will just have aĀ line like this just give a quick sketch andĀ Ā
then of course you know the line will keep onĀ going down forever and keep going up foreverĀ Ā
now here's the deal X cannot be equal to 3 weĀ will also have to put that into consideration onĀ Ā
the graph so let's say three is right here becauseĀ earlier it was up one to the right four times yepĀ Ā
now we are going to go up and then carry off thatĀ point on the graph so we will have an open circleĀ Ā
here and here is the key we have to find out theĀ Y value of this open circle because that's the YĀ Ā
value that's missing and what can be anything elseĀ except for that so how can we do it well in thisĀ Ā
case that's the concept of the limit we are goingĀ to study the behavior of the function around 3.Ā Ā
what we can do is actually just go ahead and put aĀ three not to the original though because otherwiseĀ Ā
we will get 0 over 0 [Music] . go ahead and justĀ put it here so if you have this right here oneĀ Ā
over four times three and then plus one over fourĀ we can work this out this right here is just goingĀ Ā
to be three over four plus one over four andĀ that will just give us 4 over 4 which is oneĀ Ā
so in fact when X is equal to 3Ā the Y value here it will be 1.Ā Ā
and that's actually the missing y value and whenĀ you have the picture you know to find the domainĀ Ā
so much better you know have a picture first rightĀ here I can tell you the range will just go into BĀ Ā
we go for negative Infinity right becauseĀ y can go all the way down up to oneĀ Ā
right once missing so stop like this and thenĀ Union we go from one to Infinity all right
that's it and perhaps I will tell you guys thisĀ right here uh there's a open circle hole at threeĀ Ā
comma one and that's why 3 is not in the domainĀ and likewise one is also nine the range that's it
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