1.8 - Finding the inverse of a rational function with a root as a denominator - coolmath - Free Educational videos for Students in K-12

1.8 - Finding the inverse of a rational function with a root as a denominator - coolmath - By Mr. McLogan's Math Channel

Transcript


00:00	So I have three different relationships here between X and
00:03	Y , and I want to think about which of
00:05	these , if any , are proportional relationships . And
00:08	then I want a graph him to see if we
00:09	can see anything visually . That makes them obviously proportional
00:13	. And just as a reminder of proportional relationship is
00:16	one where the ratio between the two variables and let's
00:18	say we took the ratio between why and acts .
00:21	You could also go the other way around the ratio
00:22	between X and Y , but the ratio between Y
00:25	X is always going to be some number , some
00:27	constant number . Or you could rewrite it another way
00:30	. If you were to multiply both sides of this
00:32	equation , Times X , you could see it in
00:34	a proportional relationship . Why is always going to be
00:37	equal to some constant times X So with that out
00:41	of the way , let's look at these three relationships
00:43	. So this one over here , let me drew
00:45	another column here . Another another column . This is
00:50	Let me call this the Y over X column .
00:52	I'm just gonna keep figuring out what this ratio is
00:54	for each of these pairs . So for this first
00:58	pair when ? When X is one . Why is
01:00	one half ? So this ratio is one half over
01:02	one ? Well , one half over one is just
01:04	the same thing as one half when X is four
01:07	, wise to this ratio is gonna be to over
01:09	four , which is the same thing as one half
01:13	when X is negative . Two . And why is
01:15	negative one ? This ratio is negative , one over
01:17	negative two , which is the same thing as one
01:20	half . So for at least these three points that
01:23	we sampled from this relationship , it looks like the
01:25	ratio between why next is always one half . In
01:29	this case , K would be one half we could
01:31	write why over X is always equal to one half
01:35	, or at least for these three points that we
01:37	sampled will say . Well , maybe it's always the
01:38	case for this relationship between X and y . Or
01:42	if you want to write it another way , you
01:43	could write that Why is equal to one half X
01:47	Now let's graph this thing well , when X is
01:50	one wise one half what excess four . Why is
01:55	too what X is negative ? Two . Why is
01:58	negative one ? I didn't put the market for negative
02:01	one would be . But right about there . And
02:03	so if we say these three points air sampled on
02:06	the entire relationship in the entire relationship is why is
02:09	equal to one half X well , the point ,
02:11	the line that represents , or the set of all
02:14	points that would represent the possible X y pairs .
02:18	It would be a line . It would be a
02:20	line that goes through the origin because look , if
02:24	x 01 half times zero is going to be equal
02:28	to why And so let's think about some of the
02:31	key characteristics one , it is a line . This
02:34	is this is a line here . It is a
02:37	linear relationship , and it also goes through the origin
02:41	. And it makes sense that it goes through an
02:43	origin because if in a proportional relationship , actually ,
02:46	when you look over here 0/0 , that's indeterminant form
02:49	, and then that gets a little bit strange .
02:51	But when you look at this right over here ,
02:52	well , if x zero and you multiplied by some
02:54	constant , why is going to need to be zero
02:56	is well , so for any proportional relationship . If
03:00	you're including when X equals zero , then Why would
03:03	need to be equal to zero as well . So
03:05	if you were , if you were to plot its
03:06	graph , it would be a line that goes through
03:09	the origin . It would be a line that goes
03:13	through the origin , and so this is a proportional
03:15	relationship and its graph is represented by a line that
03:18	goes through the origin . Let's look at this one
03:21	over here , this one in blue . Let's think
03:23	about whether it is proportional . We could do the
03:25	same test by calculating the ratio between Y and X
03:29	. Why and X So it's going to be C
03:33	for this . First one's gonna be 3/1 , which
03:35	is just three . That's gonna be five over to
03:39	five over too . Well , 5/2 is not the
03:42	same thing as three . So already we know that
03:45	this is not proportional , not not proportional . This
03:51	is not that we don't have to look . We
03:52	don't even have to look at this third point right
03:54	over here , where if we took the ratio between
03:57	why and access negative one over negative one , which
03:59	would just be one . Let's see , let's graph
04:01	this just for fun is what it looks like when
04:03	X is one wise three when x is one wise
04:07	three . What access to wise five Access to why
04:12	is five and when X is negative one . Why
04:17	is negative one What X is negative one ? Why
04:21	is negative one ? I forgot to put the hash
04:24	mark right there was right around there . And so
04:26	if we said Okay , let's just give the benefit
04:28	of doubt that maybe this is a these air three
04:31	points from a line because it looks like I can
04:33	actually connect them with the line . Then the line
04:35	would look something like this that the line would look
04:39	something like the line would look something like this .
04:44	So notice this is linear . This is , ah
04:47	, line right over here , but it does not
04:50	go through the origin . So if you're just looking
04:52	at a relationship , visually , linear is good ,
04:56	but it needs to go through the origin as well
04:58	for it to be a proportional relationship . And you
05:00	see that right here ? This is a linear relationship
05:02	, or at least thes . Three pairs could be
05:04	sampled from a linear relationship , but that but it
05:07	does . The graph does not go through the origin
05:10	and we see here when we looked at the ratio
05:12	that it was indeed not proportional . So this is
05:14	not proportional . Now let's look at this one over
05:17	here . Let's let's look , let's look at what
05:20	we have here . So I'll look at the ratios
05:25	y over X . So for this first pair ,
05:28	1/1 , then we have four over too . Well
05:31	, we immediately see that we are not proportional and
05:34	then 9/3 . It would be three . So clearly
05:37	this this is not a constant number here . We
05:39	don't always have the same value here . So this
05:40	is also not not proportional , not proportional . But
05:47	let's graph it just for fun when X is one
05:50	, Why is one when X is too ? Why
05:53	is for this ? Looks actually looks like the graph
05:55	of this looks like the graph of why is equal
05:58	to X squared when X is three wise nine .
06:02	At least these three points are consistent with it .
06:04	So 123456789 So it's gonna look something . And so
06:12	if this really is , if these points air sampled
06:14	from y equals X squared , then when x zero
06:17	, Why would be zero , so this one actually
06:19	would go through the origin . But notice it's not
06:21	a line . It's not a linear relationship . This
06:25	right over here is the graph of y equals X
06:28	squared . So this one also is not proportional .
06:31	So once again , these three points could be sampled
06:34	from y equals one half X . These three points
06:37	these three points could be sampled from We'll see why
06:40	is equal to , Let's See . It looks like
06:42	a line when this looks like it could be .
06:45	Why is equal to two X plus one ? So
06:49	it's a linear relationship , but it does not go
06:51	through the origin , so it's not proportional . And
06:54	this these three points looks like it could be looked
06:56	like they could be sampled from y equals X squared
06:58	, which goes through the origin when x zero .
07:00	Why is zero ? But it's not a linear relationship
07:04	, but anyway , you look at it . If
07:06	you look at it visually , has to be a
07:07	line that goes through the origin . Or , if
07:09	you look at a table of values , you just
07:10	look at the ratios and the ratios always have to
07:13	be the same value . And that was on Lee
07:15	the case with this magenta one right over here

Summarizer

DESCRIPTION:

In this video tutorial I will show you how to find the inverse of a function f(x). I do this by first writing the equation substituting y for f(x). I then solve the equation for x. The result is the inverse. The inverse can be written, substituting y back in for x, and x for y. I also show how to verify that a function is an inverse of another function. I am a math teacher that provides free online math tutoring by working through problems for the website www.freemathvideos.com These questions are usually to help students understand their math homework or to do well on their math test. The videos are made for students that want help with math and want to be shown a step by step process into solving the problems. In addition to solving the problems I also offer explanations into why I am doing each step. I offer free math videos so that my students can learn math online if they missed a day of school. However I also like to answer questions that are no

OVERVIEW:

1.8 - Finding the inverse of a rational function with a root as a denominator - coolmath is a free educational video by Mr. McLogan's Math Channel.It helps students in grades 9,10,11,12 practice the following standards HSF.BF.B.4.a.

This page not only allows students and teachers view 1.8 - Finding the inverse of a rational function with a root as a denominator - coolmath but also find engaging Sample Questions, Apps, Pins, Worksheets, Books related to the following topics.

1. HSF.BF.B.4.a : Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.