What Is a Function Precalculus

what is function have you ever wondered the answer to that question what is function function is relationship it's special kind of relationship between the input and the output or you could say between two sets of numbers it relates each input to single output so here are some examples of functions is function is function of and so is but most commonly you'll see of as function is equal to of so of could be anything of could be 3x plus 5 that's function could be 2x minus 7 that's linear function these are all examples of functions now going back to this one is the input of the function and is the output of the function now in order for this to be function for every input there has to be single output if you get two outputs for one input it is not function it's relation so in order for relation to be function there has to be one output for every input the input represents the domain of the function whereas the output represents the range of the function now let's talk about how we can make function table let's say is equal to 2x plus 4. if we plug in one or if we replace with one we'll see that is equal to six if we replace with two we get value of 8. so for each different value we get different value that's an example of function when is 3 is 10. now consider these three function tables which one is not function take minute and analyze those three tables feel free to pause the video so looking at the first table all of the values are different so that's going to be function now you can have the same value for different values that is totally okay you still have function here for the second table we do have common value but they lead to the same value which is okay for the third table we have common value but they lead to two different values that is not okay when is two can't be seven and eight at the same time and it's it needs to give us just one output not two different outputs so if we input two we want the function to give us one output if it gives you two outputs it is not function it's relation but it's not function consider these six graphs which one represents function and which one does not represent function go ahead and identify each one as function and you know the ones that are not functions feel free to take minute on this one so how can we tell if graph is function or not there is something called vertical line test so if you draw vertical line if it touches the vertical line at one point it's function if it touches it at two or more points then it is not function so or the graph that is associated with is function the graph that is associated with that is not function because it touches the vertical line at two points so let's put for not function now for that's function it touched it at one point graph does not pass the vertical line test so that is not function if you draw vertical line from anywhere on it only touches it at one point so that's the function and for we can clearly see that it's not function at this point it touched it at three points when we place the vertical line at that location so is definitely not function by the way for those of you who want more examples on functions evaluating functions dealing with composite functions and like operations associative functions check out the links in the description section below i'm gonna be posting more video content there for those of you who needs to go deeper into this subject but now let's talk about the different types of functions that you'll encounter in typical algebra or even pre-calculus course the first one you need to be familiar with is this one this is linear function as you can see it's straight line and the parent graph is equals now you can have variations of that for example equals 2x is linear function or equals 3x plus 7 that's the linear function is raised to the first power the next level is quadratic function which looks like parabola and the parent function is is equal to squared so you could have is equal to squared plus four that's quadratic function or is equal to three squared plus seven that's quadratic or is equal to squared minus plus eight that's quadratic function too the leading term is an squared you could have number in front of it or you may not but that is an example of quadratic function now let's look at the next example this is called cubic function and the parent graph for that is is equal to cubed next we have this one this is is equal to the square root of and it's radical function you can also call it square root function and then there's one that starts from the y-axis and becomes horizontal this is logarithmic function and the general form is is equal to log so that function increases at decreasing rate and there's another function that increases at an increase in rate that is called an exponential function it's increasing at an exponential rate and this would be like is equal to raised to the or you could have another constant like is equal to 3 raised to the is just number it's approximately 2.718 and there's some other numbers that come after that so that is an example of an exponential function next you have trigonometric functions and these go on forever in horizontal direction so that is an example of is equal to sine so that's trigonometric function these are periodic functions because they continue forever they repeat the same pattern now we do have some more functions to review the next one is an absolute value function i'm sure you've seen this one before and the graph is where the function is is equal to the absolute value of next we have rational functions this is where the degree of the denominator is greater than degree of the numerator so good example of rational function is is equal to one over so typically you'll see very let me say that again you'll see variable on the bottom of the fraction and then there are polynomial functions these are functions with many terms and the shape of polynomial function can vary it won't always look like that but this would be something like is equal to to the fourth minus cubed plus two squared minus seven now keep in mind that is just an example this specific equation doesn't represent that particular graph is just an example of polynomial function so those are some different types of functions that you're going to encounter in typical algebra course there are some other functions but we're going to stop it there now you need to be able to evaluate functions for instance let's say that of is squared plus 4x minus 7. what is the value of of 3 how would you find the answer to that question for this all you need to do is simply plug in the value it's going to be 3 squared plus 4 times 3 minus 7. so that's going to be plus 12 minus 7 12 minus 7 is 5 and 9 plus 5 is 14. so that's the value of of 3. but sometimes you may have multivariable function like this one let's say that of comma is 2x squared minus squared plus 3xy so in that case what is the value of comma 2 mean two comma three in this case just replace with two and replace with three and then you're gonna get the answer so two squared is four times two is eight three squared is nine three times two is six times three that's eighteen eight minus nine is negative one negative one plus eighteen is seventeen so that's how you can evaluate multivariable function so that's it for functions if you want more examples and other harder problems check out the links in the description section below