Functions and Graphs - Free Educational videos for Students in K-12

Functions and Graphs - By The Organic Chemistry Tutor

Transcript


00:00	So this video is a multiple choice review of functions
00:03	and graphs . Let's start with the first problem .
00:06	Consider the function F of X is equal to x
00:09	squared minus five X plus seven . Which of the
00:12	following is equal to F three . So to evaluate
00:15	the function , We simply need to replace acts for
00:18	three . So it's gonna be three squared minus five
00:21	times three plus seven , three squared or three times
00:25	street . That's nine . five times 3 is 15
00:29	and nine minus 15 is negative six . Negative six
00:33	plus seven Is equal to one . So therefore f
00:37	of three Is equal to one . And that's the
00:40	answer , which correlates to answer choice C number two
00:47	. If F of X equals 10 , which of
00:50	the following could be a value of X . By
00:53	the way , I recommend pausing the video and working
00:55	out the problem yourself . If you do so you're
00:57	going to get a lot more out of this lesson
01:01	. Now , since F of X is equal to
01:03	10 , We can replace f of X with 10
01:08	. Our goal in this problem is to solve for
01:10	the value of X . So let's begin by adding
01:12	eight to both sides , negative eight plus eight is
01:16	zero , which is nothing 10 plus eight is 18
01:23	. So now when you divide both sides by two
01:26	, two divided by two is 1 , 18 ,
01:28	divided by two is 9 , so nine is equal
01:31	to the absolute value Of X -5 . Now to
01:35	get rid of the absolute value symbol , I need
01:38	to write two equations . nine is equal to X
01:42	-5 and -9 is equal to X -5 . So
01:46	we're gonna get two possible answers first , let's add
01:51	five to both sides . Nine plus five is 14
01:57	and then let's do the same to the other side
01:59	or the other equation -9 Plus five is -4 .
02:05	So we have two possible values for X . X
02:09	can equal 14 or it can equal negative four .
02:12	However , -4 is the only one listed . So
02:14	therefore be is the right answer number three . What
02:20	are the following ? Is a function ? Is it
02:23	a B , C or D . Now for a
02:28	graph to represent a function , it has to pass
02:30	the vertical line tests . So let's look at answer
02:33	choice A . If we draw a vertical line ,
02:36	Notice that it touched it at two points . Therefore
02:39	A does not represent function . Looking at answer choice
02:43	, see it touches the vertical line at three points
02:46	so it does not pass the vertical line test .
02:48	So C . Is not a function for a curve
02:52	to pass the vertical line test , it must touch
02:55	the vertical line only at one point . So for
02:58	answer choice D . It touches the vertical line at
03:01	two points . So D does not represent a function
03:04	, But for be it only touches it at one
03:06	point . So answer choice B is a function number
03:12	four . What is the value of f of -1
03:16	? According to the graph shown below . So how
03:20	can we determine the value of f of negative 1
03:25	? Now it's important to understand that when you're dealing
03:27	with functions X is the number inside the function and
03:31	the entire function is equal to Y . So we're
03:34	looking for the value of why when Acts is equal
03:36	to negative one . Acts is equal to negative on
03:39	at this point . So me to find a curve
03:42	And we can see that why is equal to two
03:47	. So when X is negative one , Y is
03:49	too . So we have the point negative one comma
03:51	two . So this is the answer we're looking for
03:56	F of -1 is equal to two , which means
04:01	that D . Is the right answer choice . Number
04:06	five , If f of X is equal to three
04:10	, which of the following could be a value of
04:12	X . Recall that we said that F of X
04:18	is equal to Y . So if f of X
04:21	is equal to three , then we can clearly see
04:24	that . Why is equal to three . So .
04:28	Ny history , what is the value of X ?
04:32	So why is story at this point ? So if
04:35	we draw a line why equal street at this point
04:39	and at this point . So now let's locate the
04:41	X values . So X can be negative too .
04:45	Or it can be some other number which is probably
04:48	approximately about five . So five is not listed as
04:52	one of the answer choices but negative to us .
04:55	So X can be negative too . Number six ,
05:01	what are the intervals where f of axes increasing ,
05:05	decreasing and constant . So let's talk about when it's
05:09	increasing . It's increasing in this section and it's also
05:15	increasing in this section . So one point of interest
05:21	is -2 and it's always increasing before that . So
05:25	that's negative infinity and it begins to increase again when
05:28	accessory . And since we have an arrow , it
05:31	continues to increase all the way to positive infinity .
05:35	So in writing intervals you're dealing with the X values
05:38	, not the Y values . So the intervals where
05:42	the function is increasing is negative infinity to negative two
05:47	. That's the first part . And then to connect
05:49	it with the second part , we need to use
05:51	the union symbol . So union three to infinity .
05:56	Now , what about when the function is decreasing ?
05:59	Its decrease in here and here ? That's when the
06:02	values of why it's going down . So that's from
06:06	negative to 2 -1 . And from 2 to 3
06:13	it's decreasing as well . So I'm going to write
06:17	negative two comma negative one , Union , 2 to
06:22	3 And then finally one . Is it constant ?
06:27	It's constant in this region . That is from negative
06:32	1 - two . And so that's it . So
06:37	we have the intervals where the function is increasing ,
06:40	decrease in and constant Number seven identified the location of
06:49	the relative maximum of F . Of X . So
06:53	the maximum , the relative maximum looks like a mountain
06:56	or hill . The relative minimum looks like a valley
07:02	. So this is the relative maximum that we're looking
07:04	for and the X value is associated with the location
07:09	of the relative extremely . So it's located at Acts
07:12	equals -2 . So therefore be is the right answer
07:18	. Number eight . What is the relative minimum value
07:21	of F . Of X ? The relative minimum is
07:25	located right here . Now from the last problem ,
07:28	we saw that the location of the extreme value or
07:32	the relative extreme value is associated with the X coordinate
07:36	. The y coordinate is associated with the value itself
07:40	, not the location . So the value of the
07:43	relative minimum Is -2 . That's the Y coordinates of
07:48	this point . And so therefore be is the right
07:51	answer number nine . What is the value of f
07:57	. A 4 ? So we have a piecewise function
08:02	. And which part of the piecewise function should we
08:05	use ? Is it x squared plus four or seven
08:08	? X minus six ? Now four is greater than
08:14	two . It's not less than a negative three .
08:16	So therefore we need to use the first one .
08:19	So it's gonna be four squared plus four . Four
08:25	squared is 16 , 16 plus four is 20 .
08:28	So if a voice equal to 20 , which means
08:31	that D . Is the correct answer choice Number 10
08:36	, what is the domain and range of the graph
08:38	shown below ? So let's start with the domain .
08:43	The domain represents the X . Values and let's express
08:47	it using interval notation . So the lowest X value
08:52	is negative five , and then the highest X value
08:56	for this portion of the graph Is -2 . Then
09:00	it starts up again positive too . And then this
09:03	arrow tells us that it goes to infinity . Now
09:08	we have a close circle , so we need to
09:11	include negative five . So the domain is going to
09:14	be negative five to negative two Now because we have
09:18	an open circuit negative to need to use a parentheses
09:21	to show that -2 is not included . So x
09:25	doesn't equal negative too , but it's less than -2
09:28	. And then union to to infinity . Always use
09:33	a parentheses symbol for infinity . Now , if you
09:37	want to write this using inequalities , you could say
09:40	that X is less than -2 , But equal to
09:46	or greater than -5 . That covers this part .
09:49	You can also say that X Is greater than or
09:54	equal to two . So we can use an or
09:58	statement . Now , what about the range ? The
10:05	lowest y value that we see the rangers associate with
10:08	the white values , the lowest one . Is that
10:11	-5 ? And then the highest one for the first
10:15	part of the graph is negative three . Then it
10:18	starts up at one and then the arrow tells us
10:20	that it goes up all the way to posit infinity
10:25	. So to write the range and interval notation ,
10:27	we're going to start with the lowest y value of
10:29	negative five And we have a close circle , so
10:32	it includes negative five And then it stops that -3
10:36	. And then we need to connect the first part
10:37	with the second part . So we're gonna use a
10:39	union symbol . It's going to start back up at
10:42	one and go all the way to infinity . So
10:45	as an inequality , we could say that why Is
10:48	less than -3 but equal to or greater than -5
10:53	. We could also say that why is equal to
10:55	or greater than one ? So now you know how
10:58	to write the domain and range ? Using inequalities and
11:03	interval notation . So that's it for this problem .
11:07	Number 11 . Find the difference quotient of the function
11:10	shown below . So here's the formula that will help
11:14	you to determine the difference . Question . It's F
11:16	of X plus H minus F of X , divided
11:20	by H . So what is F of X plus
11:23	H To determine F of X plus H simply replace
11:28	acts with X plus H . So F of X
11:34	plus H is going to be the square root of
11:36	X plus H plus two . And F of X
11:39	itself is just a square root of X plus two
11:42	. So how can we simplify this expression ? So
11:46	what we need to do is multiply the fraction by
11:49	the conjugation of the numerator . So the conjugal is
11:53	going to be the square root of X plus H
11:55	plus two , but instead of minus , it's going
11:58	to have the opposite sign plus and then square exports
12:02	to . Now , whatever you do to the top
12:05	of a fraction , you must also due to the
12:07	bottom of fraction so that the value of the fraction
12:11	stays the same . So now we need to foil
12:17	the square root of X plus H plus two times
12:20	itself . The square root will cancel and it will
12:24	give us the stuff on the inside , X plus
12:26	H plus two . Now if we multiply these two
12:30	terms that's going to give us plus square root X
12:35	plus H plus two times the square root of X
12:39	plus two . And then if we multiply these two
12:42	terms we're gonna get the same thing but positive and
12:46	this one should have been negative . So let me
12:47	just change that due to this negative sign . And
13:03	then if we multiply the square root of X plus
13:05	two times the square root of X plus two with
13:09	the negative sign that's going to be negative acts plus
13:12	two . And so this is all divided by H
13:17	times that stuff . Mhm . Now let's see what
13:29	we can cancel . So these two terms will cancel
13:33	the adults . Zero . Yeah . And so what
13:39	we have left over it's going to be X plus
13:42	H plus two . And then we need to distribute
13:46	the negative science that can be negative ex excuse me
13:49	Negative X -2 . Yeah . And on the bottom
13:53	it's going to be aged times the square root of
13:55	X plus H plus two plus this . So now
14:00	we can cancel the 22 plus negative 20 and we
14:04	can cancel X . So we're left with H divided
14:10	by H times the same stuff . So now H
14:20	divided by HS one . So the final answer for
14:23	this problem . The difference question It's equal to one
14:27	divided by the square root of X plus age plus
14:31	two plus the square root of X plus two .
14:35	So this is the final answer .

Summarizer

DESCRIPTION:

This precalculus provides a basic introduction into functions and graphs. It contains plenty of examples and multiple choice practice problems.

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Functions and Graphs is a free educational video by The Organic Chemistry Tutor.

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