What is the range of the following relation?
1 answer:
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Before we find 
, let's think for a minute about what 
does, and what it means to find the inverse of a function. A function essentially takes a value <em />
from the <em>domain</em> and maps it to another value <em />
in that function's <em>range</em> according to a set of rules. Here, those rules are defined by the formula
All the inverse does is <em>swap the domain and range of the function</em>. Now, instead of trying to map to , we're trying to find a set of rules that'll map back to . To find those rules, all we have to do is solve the above equation for .
First, we'll multiply both sides of the equation by
to get it out of the denominator:
Cancelling on the right side and distributing on the left, we get:
Next, we collect all of our x terms on one side, and all of our non-x terms on the other:
Let's rearrange the right side and factor out an x on the left:
And finally, we divide both sides by
to obtain our answer:
![[x(3y-4)]/(3y-4)=(y+2)/(3y-4)\\\\x= \frac{y+2}{3y-4}](https://tex.z-dn.net/?f=%5Bx%283y-4%29%5D%2F%283y-4%29%3D%28y%2B2%29%2F%283y-4%29%5C%5C%5C%5Cx%3D%20%5Cfrac%7By%2B2%7D%7B3y-4%7D%20)
This equation gives us the "rules" for mapping any given y in the range back to an x in the domain. If we swap the domain and the range, we can define our function
All the inverse does is <em>swap the domain and range of the function</em>. Now, instead of trying to map
First, we'll multiply both sides of the equation by
Cancelling on the right side and distributing on the left, we get:
Next, we collect all of our x terms on one side, and all of our non-x terms on the other:
Let's rearrange the right side and factor out an x on the left:
And finally, we divide both sides by
This equation gives us the "rules" for mapping any given y in the range back to an x in the domain. If we swap the domain and the range, we can define our function
Step-by-step explanation: To draw a line plot for the data shown here, first begin by drawing a line.
Next, notice that the smallest number in our data set is 10 so we begin our line plot at 10. Next, notice that the greatest number in our data set is 22 so we continue our line plot until 22.
Now we fill in the numbers in between.
So we have 10 which we already have, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, and 22.
Now, notice that there is one 10 in our data set so we put one x above 10. Next, notice that are no 11's, no 12's, and no 13's in our data set. So we put no x's above 11, no x's above 12, and no x's above 13. Next, notice that there is one 14 in our data set so we put one x above 14. Now, notice that there is one 15 and one 16 in our data set so we put one x above 15 and one above 16. Since there are no 17's or 18's in our data set, we put no x's above 17 and no x's above 18. Finally, notice that there is three 19's, no 20's, one 21, and one 22 in our data set so we put three x's above 19, and one x above 21 and 22. We put no x's above 20.
I have attached my line plot in the image provided.
