this section we're going to talk about what makes a function so starting off what's a relation a relation is any set of ordered pairs so basically anything that can be graphed is a relation the first component of the ordered pair is the domain so that means the domain are the x values and the second components are the range that means the range is the y values so name the domain and range here when we do this we want to make sure that our numbers are in order and we start and end our list with braces so the domain are the x values so we've got 0 10 20 30 and 40. those are in order so we're going to go ahead and list them here close it with the brace and then the range is our list of the y values we want to put them in order again so we need to start with 6.7 then 9.1 10.7 13.2 and 21.2 a function is a correspondence from a first set to the second set such that each element in the domain corresponds to exactly one element in the range okay what does this mean simply this means that no x values can be repeated okay so make a make a note of that on your notes no x values can be repeated so if we look at these two we can see if we look at the x values of the first one 1 3 6 and 8 they're all different so that means that this graph is a function if we look at the x values of part b 2 2 and 3 2 is repeated which means we do not have a function if an equation is solved for y and more than one value of y can be obtained then the equation does not qualify as a function of x so let's look at an example of what i'm talking about here determine whether the equation defines y as a function of x so solve it for y is the first step subtract x squared from both sides and take the square root of both sides when we do that we have a plus or minus the plus or minus means that we could have more than one x value for the y value so therefore this is not a function function notation let's say we have a graph that says y equals x plus 2 okay function notation is just rewriting this with an f of x in place of the y f of x and y are the same thing okay so either way you see it written it means the same thing but we write it as f of x just to help us when when looking at different problems okay it is a that is the function notation and f of x what that means is what is the y value when x is whatever okay that's what that means so looking at an example f of negative 5 means what is the y value when x is negative five so to solve this we plug in negative five for x make sure here when you're squaring x you're squaring all of negative 5. so you have to make sure that that negative 5 is in parentheses because negative 5 quantity squared is negative 5 times negative 5 which is positive 25 big deal [Music] negative 2 times negative 5 is 10 add those together and we get 42. so what this means again is when the x value is negative 5 the y value is 42. so the point negative 542 would be on this graph here we're finding f of 3a well i realized that you wouldn't have an x value of 3a but this prepares us for things in the future so to find f of 3a that's where the x value is 3a we plug in 3a for x again we are squaring all of 3a so make sure you put that in parentheses 3a squared is 3a times 3a which is 9 a squared 2 times 3a is 6a plus 7. and that's as far as we can go with that it's okay that we didn't end up with a an integer with a number that's as far as we can go that's your solution for this one what is when we plug in x of 4 plus z so again make sure that 4 plus z is in parentheses because we're squaring all of it make sure it's in parentheses here because we're distributing the negative 2 to both the 4 and the z so 4 plus z squared remember that's the same thing as 4 plus z times 4 plus z i won't always write that out on the notes but i will for now so you would fool this first 4 times 4 is 16 outer 4 times z is 4z enter z times 4 is 4z last z times z is z squared then going back up to negative two times four plus z distribute the negative two so negative two times four is negative eight negative two times z is negative 2z we'll go ahead and tag that on down here combine our like terms so we've got a 4z a 4z and a negative 2z we've got just 1 z squared so i'm going to start with the z squared and then combine our z's 4z plus 4z is 8z minus 2z that gives us 6z and then we have 16 minus 8 plus 7. so 16 minus eight is eight and eight plus seven is fifteen okay the graph of a function the graph of its ordered pairs so for example five it wants us to graph the function two x and two x minus three and the same rectangular coordinate system so let's do that 2x let's we're just going to pick x values and plug them in and figure out what we get for the y's so 2 times x two times negative two is negative four two times negative one is negative two two times zero is zero two times one is two two times two is four so if we graph this we've got negative 2 negative 4 negative 1 negative 2 0 0 1 2 and 2 4. and so this function is a line that looks like this the second one g of x is equal to two x minus three so we plug in our x values again two times negative two is negative four negative four minus three is negative seven two times negative one is negative two negative two minus three is negative five two times zero is zero zero minus three is negative three two times one is two two minus three is negative one and two times two is four four minus three is one and so we graph these the first one we can't fit on our graph that's okay then we've got negative one negative five 0 negative 3 1 negative 1 and 2 1. and so we've got another line there is g of x next how do you look at a graph and determine different things from the graph okay how do you determine the domain and range from the graph well again still the domain and range the domain are still the x values and the range are still the y values so if we look at this graph and we want to determine the domain then you're going to start with the x value of negative 2 and go to an x value of 1. notice these dots here are solid dots so that means that we are including them so in interval notation the domain is from negative 2 to 1. and we use brackets because that is including the x value of negative 2 and of 1 because of those solid dots then the range are the y values so starting from the bottom moving up we are starting with a y value of 0 and moving up to a y value of 3. again because of the solid dots they're included so we've got bracket 0 3 close bracket for example 7 the domain again are the x values start with the left move to the right so here we're starting with negative 2 and moving to the right to 1 except in this case at the negative 2 there's an open dot so what we have to do is we have a parenthesis before the negative 2 and then there's a closed dot where we're at 1 so a bracket there and then the y values start at negative 1 and go up to two so we're including the negative one because that's a closed dot so bracket negative one since the two is at an open dot we use a parenthesis after it and then how do you find x and y-intercepts from a graph we've done this before kind of okay so identify the x and y intercepts for this graph the x-intercepts are where it crosses the x-axis so here's one at negative 3 0 here's another one at negative 1 0 and then the last is over here at 2 0. and then what is the y-intercept it crosses crosses down here at negative 6 so the y-intercept would be at the point 0 negative 6. that concludes this section