 This is a video about functions first. We'll begin with a relation which is a collection or set of ordered pairs and Then a function now from x to y is a relation that associates exactly one element of y and take each element of x To break this down a little bit. This just means that x values do not repeat amongst the ordered pairs that are given Function notation uses f of x, g of x, h of x in place of y So don't let the notation of function scare you The domain of a function includes the x values or input values The range of a function includes the y values or output values. So x is input y is typically output All right an example one I give you a group of ordered pairs and I want to know if each set of ordered pairs is a function if so We will state the domain and range Looking at part a they give me four ordered pairs But notice one four and one seven each have an x coordinate of one So one repeats as an x value So that means it is not a function that's what we mean by saying x has to be unique part b Look at the x values negative one zero one and two the x values do not repeat. So this is a function Since it is a function. We are going to state the domain and we will state the range Since there's only four points We will just list the four x values from the points as the domain Remember the domain would include x values only and then the range or the y values you have three Five Two and then three again So include that within set bracket check set brackets 352 this is actually written more nicely as 235 Ordering the numbers is always nice Next if we're looking at a graph we can determine if it's a function using the vertical line test This means that if we walk across the function or go across the function with the vertical line We cannot touch more than one point at the same time For instance in part a with the circle I touch multiple points vertically at the same time in many places For instance, I have negative two negative one And then I also have negative two five X is not unique. That's why the vertical line test works So the answer is no for part a and for part B if you go across the graph with the vertical line Well, you never touch more than one point at the same time vertically. So part B is yes That's what you call a quadratic function actually, but the important thing is it's a function Next all lines are functions except for vertical ones. Those are of the form x equals some number So in example three Part a you have x equals seven. This is a vertical line Because it's of the form x equals a number. So the answer is not a function Part B. It is not a vertical line. So it is a function Y equals 3x plus 5 is a function Finding the domain and range of each function Or of each graph we should say because part A is not a function Remember the domain would be x values. The range would be y values We want to give the answer in ways known as interval notation So here's the trick to find the domain or x values, which the graph is defined You look at the furthest left point of the graph What's the x value? Negative five and look at the furthest right point of the graph What's the x value one so negative five comma one enclosed within brackets? Because you have points at negative five and you have a point at one That would be the domain the range. How low do you go in terms of y values? Remember range you need to be in a y mode. So negative one is the lowest you go vertically And the highest you go would be a y coordinate of five So we enclose within brackets because there are actual points to find that those y values Next part B Remember the domain would be the x values the range would be the y values So how far left through good well, let's look at part B We have arrows this left side arrow has both leftward and upward direction The right arrow has both upward and rightward direction So the furthest left you go well since there's an arrow going left. It's negative infinity the furthest right you go Since there's an arrow going right. It's positive infinity and you always put parentheses around infinities range How low do you go? Well the lowest point has a y coordinate of negative three and how high do you go? Well arrows with upward direction mean that you go all the way up to infinity Parenthesis around infinity bracket around negative three because there is a point at the y value of negative three so that's finding the domain and range of Graphs one of which was a function and one of which was not Next we can evaluate functions at x values For instance example five f of x equals three x squared plus four But they want you to find f of two that just means in the equation let x equal two So this means I have three times two squared plus four That's three order of operations two squared is four plus four Multiply three times four is 12 plus four You end up getting 16 Part B f of negative one be careful here because you're squaring x that means you have three times x squared or three times negative one squared plus four That means I have three times negative one to the second powers one plus four Three times one is three plus four Which is seven Example six same process except we have g of x equals two x minus one So when you see g of one it means in the function g let x equal one So g of one is well two x minus one that's two times one minus one That's two minus one. That is one part B. You're gonna let x equal negative four G of negative four. That's two times negative four minus one That's negative eight minus one That's negative nine. So that's how you find function values and last but not least If you plug a certain value Into a function and then you get a certain value out Well, that value you input is the x coordinate and that value you got out of the function It's called your output value for instance if f of three equals seven then three seven is on the graph of the function f So for instance example seven write each of the following as an ordered pair part a f of 12 equals negative one Get this means that your input is 12 And your output is negative one The ordered pair would be 12 negative one. That is all Part b h of negative three equals six. The ordered pair would be well inputs negative three output six So not much work to show there at all But it's an important concept that what's in the parentheses is input and then the number on the other side of the equal sign As output. Well, I hope you enjoyed. Thanks for watching