2.4 Inverse Functions

Added: 2026-05-09

[00:00:06]okay section 2.4 inverse functions okay so we're going to talk about inverse function notation we have talked about function notation but now we're going to talk about inverse function notation so let's take a look at the first example it says recall the phone example from earlier well we didn't have a phone example earlier but where the calling plan charged us 30 a monthly service fee and 10 cents per minute for long distance calls obviously this is a little old because we used to have to pay for long distance calls now you guys have cell phones so that's no longer the case but we're gonna work on this problem anyway so if you look at this t this t is the time in minutes that your input c is your output so c our output is equal to f of t which is our output again t is the output f of t i'm sorry t is the input f of t is the output so we've been we've been already talking about that but there's another notation which is the inverse notation if i take the inverse of c that is asking me to find the t okay so this notation means find the input this is the output this is the input okay this represents input so in this case we say the cost c is a function of time t this here is saying time t written as a function of the cost in this case c okay so a little bit different so let's take a look at this this is a review here f of 36 f of 36 input or output that's output 36 is your input so t is your domain so 36 the output is 33.60 now what does that mean interpret that statement [Music] so 36 minutes it's saying if you talk for 36 minutes on long distance if you talk for 36 minutes of long distance your phone bill is 33.60 okay but let's look at this notation does this mean they want to know what the output is so is 36 our input or is 36 our output so in this case it's the inverse notation inverse notation so 36 is our output we're thinking backwards now so 36 is our output our input is 60. so what does that represent well again this is what we're being given so it's saying that the opposite of this here it's saying if your phone bill is 36 then you talked for 60 minutes of long distance okay so they're inverses of each other okay you try the next one if this is given here what goes here i'm going to give you a hint another way of looking at this this inverse is blank of f 33 this means what does f of 33 equal and the answer is 33.30 33.30 so again this is the given so what is it saying saying you talked for 33 minutes long distance so your phone bill is 33 dollars and 30 cents okay so function notation and inverse function notation you need to know both okay so let's try some some problems let's look at the next page here okay let's try the first one together a find f of 2.

[00:08:03]that's saying that 2 is our input f 2 is the output so what's our output right about there well these are going by halves so it's not quite 1.5 so let's just say approximately 1.3 okay now you go ahead and try this one what is this saying is this 2 our input like it was over here or is this to our output if we're thinking backwards so this 2 is our output so we go to where y is equal to 2 on the graph and we give the x value so again it's going by halves so that's three and a half okay stop the video right now and try see through i on your own okay hopefully you try those on your own so you can get some practice here are the answers for you to check c is seven d is negative four e is approximately 2.7 f approximately 5.4 g again we're approximating about 2.4 not quite 2.5 but close enough okay h h is equal to 5 and i again i is saying x is equal to g of zero okay x is equal to g of zero now when i say x i just mean your answer this answer right here don't get confused this is saying what is your output when x is equal to zero so don't let that one's a little confusing so what do you get for that what is g of 0 okay well we're looking at this table for i i'm sorry for g this was f this is g so make sure you're using the right ones hopefully you're using the right ones if not stop the video and go back f and g so if we go to this table right here g of zero the answer they want is seven okay f you use the f graph g you use the g table so if you have to stop the video and go back and try those again if not if you have any questions make sure you ask okay let's go ahead and look at the story problems number two it says a equals f of n so that means n is your input f of n or the amount of periwinkle blue in gallons is your output so right here this input it says needed to paint n square feet of our house so what is this telling us this is saying that we have 20 square feet of house to paint so what does this function notation represent the amount of what it's the amount of periwinkle blue paint we'll just say of paint in gallons needed to paint 20 square feet of house okay now see the difference here what does this one mean what is this 20.

[00:13:06]okay is 20 our input or is 20 the amount of paint 20 you gotta think backwards 20 is the amount of paint we need okay so what is or i should say the amount of paint we have 20 is the amount of paint we have we have 20 gallons of paint the question is what does this notation mean what does it mean f of negative 1 of 20 now don't do any mathematical operations with that negative 1.

[00:13:46]that is just inverse function notation don't be multiplying by negative 1 or dividing by negative 1 or flipping it over or anything like that this is inverse function notation so normally this whole thing represents our output but this notation here represents what that's what they're asking us for it represents the number of square feet of house that can be painted with 20 gallons of paint okay all right last example here it says if cricket chirps are times per minute then the outside temperature is given by this formula here so r is our input f of r or temperature is our output find a formula for the inverse function find a formula for the inverse function of t okay so let's start with the given formula let me write that down if t equals 1 4 r plus 40.

[00:15:44]what is r no we just have a math problem here oops 40 not 400 sorry about that so if you're trying to solve for r what would you do first well i would subtract 40. so t minus 40 equals 1 4 r to solve for r r is equal to 4 times the quantity you're multiplying the entire side by 4 to get r by itself or r equals 4 t minus 160.

[00:16:37]so this is t in terms of r and this is r in terms of t okay well if this up here is f of r okay this right here when you rearrange the equation for r that's going to be your inverse notation so t here is the variable in this equation so the other notation the inverse function notation would be the inverse of t equals 4t minus 60.

[00:17:28]inverse function notation okay okay so now that we have those formulas let's see if we can calculate and interpret these let's do this one this is again what we've already been doing find f of 50. find f of 50. well we're going to use our f of our equation okay f of r equals 1 4 r plus 40. okay this 50 is our input into this equation so 1 4 of 50. plus 40.

[00:18:19]50 divided by 4 plus 40 that is 52 and a half so what did you just calculate we were given r equals 50. what is r what is r okay so r is the times per minute the cricket chirps so [Music] when a cricket chirps 50 times per minute the outside temperature is 52.5 degrees degrees fahrenheit okay but what does this notation mean that when this is the given so this f of negative 1 of 50 we're going to use this equation okay now you could use this equation but is this the r or is this the t well it's the t so if you use this equation it you'd get the right answer but since we already have this solved for t we can use that one okay so inverse of t equals 4 t minus 160.

[00:20:44]so that means that when the temperature is 50 degrees fahrenheit we end up getting uh 200 minus 160 we end up getting 40. 40 what 40 chirps per minute so interpret what's given so again this is temperature so this one is saying when the temperature is 50 degrees fahrenheit just to be clear a cricut will chirp 40 times per minute now i don't know about you but i've got a lot of crickets around my house so maybe somebody can see if this actually this formula actually holds true