section 1.2 titled functions and their properties we'll begin this section by defining a function uh talking about what a function is how can we determine if what we have is a function uh we're going to discuss the domain and range of functions and then we're going to talk about different types uh different sort of characteristics functions have um continuity extreme values symmetry ASM tootes um in behavior um a whole lot of different things we're going to go over just kind of brushing the surface on uh in this particular section uh all things that we'll come back to in more depth later on in the course so first what is a function well we can define a function to be this so a function is a relation in which each element of the domain maps to exactly one element of the range in other words every X can have only one y each X gets one y simple enough so how about we look at a mapping diagram of a function and not a function just to kind of compare and contrast the differences between the two um so let's say over here the example that is a function let's say that this x value maps to that y value in other words X1 comma y1 is a point on the function let's say X2 m maps to Y2 let's say X3 maps to Y3 and let's say X4 maps to Y3 that is a function because every X over here goes to only one y it doesn't matter that these two x's here point to the same Y what matters is that this x points to one y value and this x points to one yval okay on the other hand let's check this one here let's just say X1 goes to y1 X2 to Y2 X3 goes to Y3 and so forth X4 goes to y4 right now as it's drawn this is a function um but let's say that X1 also goes to Y3 this is not a function now now because X1 has two y values okay so every X has one y it's a function if a single X has more than one y it's not a function now a mapping diagram is just one way to determine if you have a function or not have a function another way is to do a vertical line test vertical line test is a nice visual test for dealing with functions what you do is you take a vertical line you say you take this vertical line here and what you're going to do is and I can't pick it up and move it but let's just use the pin here uh you take this vertical line and you sweep it back and forth through the graph okay um in order for it to be a function the vertical line can never hit the graph more than once okay um this vertical line would never touch the graph more than once as it goes from left to right okay so this is a function on the other hand if I were to take a vertical line over here and I sweep it through left and right we you can see there are a lot of places on this particular graph where the vertical line hits the graph more than once it's right here it's right there it's right here so what we can see is this particular x value maps to a y1 a Y2 a Y3 this particular X has three y values so this is not a function next we're going to talk about domain and range now if you remember back to one of the appendix sections uh we talked about domain of algebraic expressions and that translates very nicely to finding domain of algebraic functions um one of the two things that we said caused a problem in domain were square roots um this particular function has a square root um now we know that over real numbers uh we cannot square root a negative number okay so um what I'm going to do is take this part right here the x + 4 which is under the square root and say that that x + 4 must be greater than or equal to zero in other words it can be negative okay um if I solve this for x x has to be greater than or equal to -4 X's our domain so we've just essentially found the domain uh the domain for this function are all X's greater than or equal to -4 or preferably -4 to Infinity in other words X cannot be anything smaller than -4 um for instance -5 for X -5 + 4 would be -1 we cannot square root the 1 okay so4 is the smallest value that we can use now for the range we're going to use a graph and right now since we haven't studied the graphs in huge detail this year um we're going to use the graphing calculator so what I did is I went to the y equal screen I typed in the Square < TK of x + 4 we hit graph and sure enough we can verify from our graph that it starts at-4 on the X and it goes bigger than -4 okay now let's use the picture to determine the range the Y values we can see that the smallest y value right here is at zero and then the graph starts to go up from there so that's going to be our range our range is zero to Infinity so typically the domain we've got algebraic ways to do it we can confirm with the graph uh the range we pretty much just use the graph all the time to do that in the next problem again finding domain and range we again begin with this square root piece on top maybe and because there's an X underneath the square root we know that X has to be greater than or equal to zero can't do negative numbers okay now there's another problem inherent with this function and that is the the fraction we have to be aware of any value that might cause us to divide by Z okay so Mental Math can solve this and everybody's capable what number minus 3 would make the denominator be zero and the answer is three and so from the denominator here we see that X cannot be three okay so this an interval notation would be 0 comma Infinity um but we have have to exclude we have to exclude the three this includes the three so I would write my domain this way I would say that it is 0 to three the rounded bracket means that we're not going to include it and three to Infinity so now we're going 0 to Infinity but we are breaking at three we're not including at three next what we'd like to do is find the range and for the range we're going to again look at a graph um I went ahead and typed in the function notice there's parentheses behind the Divide symbol so that the X and the minus 3 are in the denominator um and when I look at this I'm looking vertically for the range um it appears that there is a y value right in the middle of these two curves that is excluded from the range and it should be pretty apparent that that y-value we're excluding is zero um so that's how I'm going to write my range is all yv values cuz it went down and it went up so it's going to be all y values except the zero so that's going to be negative Infinity to 0 and 0 to positive Infinity other words it goes below the x- axis goes above the x-axis but it doesn't ever touch the x-axis it never is zero I'd like to do one more domain and range problem and and one that's just kind of an interesting problem we said that domain can be restricted by by really two things that cause concern one are square roots the other is uh division um there is another thing that we need to look for and that is typically um when we're doing application modeling type problems um there's some times a domain inherent in the problem that we have to consider U for instance if you look at this particular function uh or this particular formula we should be able to recognize that this formula finds the volume of a sphere now notice the so here the r is like the X it's the independent variable the V is like the Y so V of R is kind of like f ofx um notice that the r here has no square root above it has no divide we're not dividing by any RS so the two cases we had prior to this wouldn't really apply here yet that still doesn't mean that R can be everything okay if we're talking about volume of a sphere okay we have a sphere and a sphere has a radius um the radius can't take on every value uh in fact the radius could only take on positive values um radius couldn't be zero if the radius was Zero we'd have no sphere and it wouldn't make any sense to have a negative radius um so sometimes domain can be restricted by the context of the problem here I would consider the domain to be zero to Infinity again can't be equal to zero because there's no sphere can't be negative because that wouldn't make any sense to have a negative length on the radius so the domain's got to be some positive number 0 to Infinity the range also has to just make sense in the context of the problem um the range the y values the Y values are the volume values we could never have a negative volume uh we couldn't have no volume so the range is going to be also zero to infinity and so this is a an example where the domain and range can be found contextually from the problem but also just using a little bit of Common Sense