Functions 2 1

追加: 2026-05-06

[00:00:06]okay here you take notes on section 2.1 we're talking about input and output so this should be a review for you again we're going to review function notation here we've already discussed this so they want us to estimate f of 10. so what does that mean this is your input 10 and f of 10 is your output so if we go over to 10 and we go up to the graph that is our output so what y value do you think that is now if you look here this is 10 i'd be going by ones or two so let's see here 2 4 6 8 10 12. yeah we're going by twos so it looks like it's almost 12.

[00:00:55]so we're gonna approximately say 11.75 okay it's just an approximation reading the graph now the next question says if f of x is equal to 10. so it's giving us the output it wants us to find the input so let's go and look where our output is 10 it happens in how many places right there right does it happen in another place actually it does it happens down here so if our output is 10 what is our input well we're going to approximate those as well so what are we going by here this is 10 is going back twos again 2 4 6 8 10 yes so this is 6 and this is 8.

[00:01:52]so it looks like this x value is a little bit bigger than seven so let's just say approximately seven let's go seven and a quarter okay it's approximation then how about this one down here approximately this would be 18 right so 19 is halfway in between it's almost 19. close enough we'll say about 19.

[00:02:22]okay maybe a little less than 19. again we're approximating okay can you answer this question find f of a find f of a here's a where's f of a it's right about there 2 4 so f of a is approximately a little more than three would you say oh no let's back up i was looking at the wrong values two four six that's trying to confuse you so f of a again here what is the output what is the y value so that's equal to six there we go okay and then the next thing find f of 10 and subtract f of 6. okay so f of 10 we have up here is approximately 11.75 so we're just going to use that approximation and we're going to subtract f of 6. have we found f of 6 yet no we haven't so let's go up here and there's six so right here is f of six okay this is eight let's see nine's right about there so why don't we go with eight and three quarters just a little bit less than nine so the difference is approximately three so just reading a graph and getting an output something that we should be able to do okay let's take a look at this example what if i gave you a table and i asked you to find f of 60 okay now again this is t that's our input and g is our output so g is equal to f of t g is the output t is the input so if 60 is the input what is the output the output is 90.

[00:04:57]but be careful include units with your answers what units do you think we need millions of tons of garbage so it's 90 millions of tons of garbage okay how about f of 7 d5 can you do that one f of 7 d5 75 is your input your t is your input g is your output so 130 millions of tons of garbage okay part c so f of t equals 165 solve means find t solve means find t okay so they're giving us the output they're giving us the output so let's find the output of 165.

[00:06:33]there's our output of 165 what is our input our input's 85 so that means t must be 85.

[00:06:45]again include the proper units 85 what 85 years after 1900 because if 60 is 1960 85 is 1985.

[00:07:20]so t is 85 means the year 1985.

[00:07:28]okay all right if you want to um turn the video off in a second and try some of these i'm going to give you um some directions when we interpret this i want you to say toward the library or we're going toward home so everybody has the same answers instead of saying away from the library we're going to stay toward home so you're either going toward the library or you're going toward home so let's try the first one it says given is the graph of the function v of t is your output t in minutes is your input it represents the velocity of a man riding his bike to the library and going back home after a little while a negative velocity so these down here a negative velocity indicates that he is riding toward his house so toward home away from the library so again we are just going to say he's going toward home we're going to just use that as our interpretation so we all have the same answers okay so let's do the first one together v of five evaluate and interpret five is our input so let's go to five what is our output what do you think that is right there well it's halfway in between 10 and 15.

[00:09:09]it's halfway in between 10 and 15.

[00:09:12]let's say 12.5 what is the unit miles per hour is the unit on our y value now interpret that answer what does that mean well if it's a positive velocity it means that he is heading toward the library okay he's heading toward the library what is this five mean after five minutes after five minutes let's try this one let's try d how about when they give us this and they want us to solve for t and interpret so again what does this represent that represents our output so let's go to the graph and let's find where our output is five our output is five here here here and here so there are four solutions t let's see what are we going by we're going by fives here so we're going to approximate what this is that's two and a half that's so we're going to go about one and a quarter 1.25 for that one if you want to stop the video and estimate the other ones you can do that let's go to this one right here just a little bit bigger than 15.

[00:11:08]we'll say that's about 16.

[00:11:11]the next one is right here just before 20 so we'll say 19 and we've got one more right around here and we will call that about 24 and those are all minutes so what is that mean what does it mean if v of t equals five it means that p is going 5 is what's given over here this 5 meant five minutes after five minutes what does this five mean this five means he was going five miles per hour and don't forget to tell me what direction was he going towards the library or toward home so because it's a positive velocity he was heading toward the library [Music] okay so now i would like you to turn off the video and try b c e and f turn off the video and try it yourself okay welcome back for 40 you should have 17 and a half miles per hour again we're approximating so that's toward home the velocity is negative you can't go negative miles per hour that just means you have a negative direction so he is going towards home after what is the 40 mean after 40 minutes okay and c you should have gotten b of 12 to be approximately seven and a half v of seven approximately 12.5 so the difference is negative five okay so how do we interpret that we have we're subtracting the two velocities and we're getting a negative answer so at one point he was going slower so after five minutes so from 12 to 7 from 12 to 7 that's five minutes and you were going five miles per hour slower after five minutes see the difference right there so you could say going five miles per hour slower after five minutes toward the library or you could say five miles per hour faster then seven and a half minutes five miles let me say that again you could also say five miles per hour faster let me write that one down i like this one better five miles per hour faster after seven and a half minutes then after 12 and a half minutes oh not seven and a half again sorry let's try that again this is your minutes 12 minutes and seven minutes so five miles per hour faster after seven we've wanted to write seven and a half after seven minutes then after 12 minutes because after seven minutes you were going 12.5 miles per hour and after 12 minutes you're going seven and a half miles per hour so you were going slower or you're going faster after seven minutes then after 12 minutes so you can phrase it either way okay let's go ahead and look at e okay if the output is negative 10 you should have come up with the t values of about 47 and 37.

[00:17:20]i'll get those backwards okay now what does it mean for a negative 10 to be your velocity negative velocity means you're going toward your house okay so you she was going going 10 miles per hour toward his house at those times at those minutes okay and then f when is v of t equal to v of ten when is v of t equal to v of ten well we have to find what v of ten is v of ten looking back up here v of 10 is right there so that's about seven and a half so it's asking you to solve that equation so when is your output seven and a half when is your output seven and a half how many answers do we have there quite a few right so for this one t is equal to 2.5 right here then we have a range from here we have about from 10 to 15.

[00:19:24]so 10 to 15 there's a range all of those in between and then right here again let's say that's about 19 and then right here that's 19.

[00:19:42]and then again right here which is [Music] you can call that i don't know probably about let's just say 23 and a half again we're approximating so those are all the times when the velocity is seven and a half so positive seven and a half means the seven and a half means you are going seven and a half miles per hour it's positive so you are going toward the library at the same velocity so those are all the times when you were traveling at the same velocity of seven and a half miles per hour towards home okay last example example number three consider the functions given we have f of x equals this quadratic there's a rational function here and they're asking us to calculate f of zero so you should be able to solve these again you can turn off the video and try solving them on your own okay let's check your answers zero is your input so your output is zero squared minus two times zero minus eight so your answer is negative eight input zero output negative eight how about this one f of x equals zero so they wanna know when your output your two your x squared minus two x minus eight is equal to zero how do we solve a quadratic well the easiest way is to do what factor x minus four times x plus two equals zero and when you set your each of your factors equal to zero you can solve for x and you get four and negative two two solutions okay let's try the next one if you haven't tried the next one turn off the video and try it yourself okay check your answer plug zero in for x that's your input and you're getting one half minus one which is negative one half okay this one is going to be a little bit harder because we have a rational function here but we're going to set our function equal to zero so we're gonna set one over x plus two minus one equal to zero the easiest way to solve a rational equation is to multiply both sides through by the denominator which is x plus two so we can go ahead and do that if we multiply this side times x plus two and we multiply this entire side times x plus two that's the easiest way to solve a rational equation so on this side we still get zero and this side we're going to distribute when you multiply x plus two times one over x plus two the x plus twos cancel and you're left with just one when you distribute the x plus two into here that's going to be multiplying it by a negative one so be careful so zero equals one distribute the negative sign and now solve for x if i move x to the other side that produces my answer of x equals negative one okay last example okay this is going to be a complex fraction this one's going to require a little bit of algebra skills so go ahead and try this one find f of one over t plus one where do i substitute this in this is our input this is our input in for x so f of one over t plus one we're gonna replace our x with this a replacement so we have an x on the top one over t plus one in the numerator we have another x in the denominator one over t plus one in the denominator plus one that's the substitution the rest is simplification of a complex fraction the easiest way to simplify a complex fraction there are many ways look at the denominator here the denominator is x plus one here you have a denominator of i'm sorry t plus one here you have a denominator of t plus one and here you have a denominator of one so the numerator and the denominators terms have a least common denominator of t plus one so what i recommend is multiplying the numerator times t plus one and multiplying the denominator times t plus one and that is a trick again just like over here rational equations we don't like fractions multiply by the common denominator same thing here denominator denominator multiply the numerator and the denominator by something equivalent to one and watch what happens when you multiply the numerators together t plus one this is t plus one over one and frozen here let me uh go back sorry hold on it's frozen i don't want it to start all the way over okay let's flip that around okay i'm not sure when i froze but let's try this again you have a substitution into the x so a replacement into the x here and here the least common denominator is t plus one so i'm going to multiply the numerator and the denominator by t plus one and watch what happens t plus one this is t plus one over one if you need to do that you can write it like that multiply these two together the t plus ones cancel and you're left with just one on the top now again you're going to distribute distribute this to here again this is t plus 1 over 1. so when you multiply those fractions together the t plus 1's cancel and you're left with one and then you're going to distribute it to this one and you're left with t plus one now we don't need the parentheses because there's a plus here so that is one over t plus two and that's your final answer you