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Transcript
MAT1193 – Notes on functions Functions are the fundamental objects that we will be dealing with in this class. A function is a procedure or rule for converting an input value into an output value: (1) put a number in, (2) turn the crank (apply the procedure/rule), (3) get a number out We often write a function with a formula, e.g. F(x) = 3*x+5, where I have used * to represent multiplication. [e.g. is a latin abbreviation meaning “for example”]. This formula means that F is the name of a procedure that takes an input number (called x for now), multiplies it by 3 and then adds 5. So F(3) = 3*3+5 = 14 (if we put the number 3 into the input, the procedure F gives us back a value of 14). The variable x is just a placeholder. We typically use (overuse) the letter x to represent an unknown variable, but we could have used any name we want. For example F(y) = 3y+5, F(@) = 3*@+5, and F(fred) = 3*fred+5 all describe the same function. Sometimes we write functions in a different way: y=3*x+5. This describes the same function (procedure) but emphasizes different things. The notation F(x)= 3*x+5 emphasizes that there is a procedure called F and that function is defined to be equal to 3 times it’s input plus 5. The notation y=3*x+5 emphasizes that there is a value y that is obtained by taking a values x multiplying by 3 and adding 5. In fact one could view a function as linking an input value to an output value. Conceptually we could view a function just as a list of of (input,output) values (x,y). The F(x)= 3*x+5 notation emphasizes the procedure and gives the function a name. The y= 3*x+5 notation emphasizes a function as a pair of input/output values and gives the output variable a name. Sometimes one way is better; sometimes the other is.
Functions can also be specified using a graph. How does a graph represent a rule for taking an input value and converting it to an output value? •
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To put a number in, go that distance on the horizontal axis (1) To apply the function, draw a vertical line and find the point of intersection with the graph (2) To get a number out, draw a horizontal line and find the length of the intersection with the vertical axis (3) In order for this procedure to give you a single output value, the vertical line drawn in step (2) must intersect the line in exactly one point (the vertical line test). The point on the graph has coordinates (x,y) or using a different notation (x,F(x)). Another way of thinking about this is that if the point (2,5) is on the graph, then by plugging in the values 2 for x and 5 for y, the equation for the function must be true. [Q: does the point (2,5) lie on the graph y=3*x+5? ANS: plug to find 5=2*2+5. This is not a true equation so the point (2,5) does NOT lie on the graph for the function y=3*x+5. Q: does it lie above or below the graph?] There are a couple of other definitions that go along with the definition of a function. The domain of a function describes the types of things represented by the variables that are put into the function machine; the range of a function represents the things that come out. In this class, we’ll be dealing with functions in one dimension: our functions take one number in and spit one number out. For the function c(z)=z2, the domain is all numbers z and the range is the positive numbers and 0 (since the square of any number is ≥0). For the function d(a) = √a, the domain is the numbers ≥0 (since you can’t take the square root of negative number); the range consists of numbers that are ≥0 as well. Note that sometimes the domain and range are determined by the problem, not just by the form of the equation. For example, suppose that t represents time from the beginning of an experiment, and w(t) represents the number of wigglies that are measured in the experiment. Also suppose that the function w(t) = t2+13 describes the number of wigglies over time. In this case the domain of the function is t≥0 because t represents the time from the beginning of the experiment and it only makes sense if that is a positive number. Functions are fundamental to this course. For most applications, the first step is to define the relevant variables, and then to formalize how those variables interact by writing down a function. For this reason, it’s very important that you think of functions as procedures and learn how to make the translation of that procedure between the language of math and the language of the real world. In algebra, the fundamental object was a variable, and solving an equation typically meant finding the value of that variable. For example, to solve 3*x+5 = 11 means finding the value of the variable x that when you plug that value into the equation you get a true statement (x=2) . In calculus, the solution to the typical problem will be a function. For example, solving the differential equation dx/dt = t2 means finding a function x(t) such that the derivative of x with respect to t (written dx/dt ) is equal to the function y(t)=t2. We’ll learn later what the derivative is and how to solve differential equations. But given that functions are the main object in this class, you need to get a gut feeling of what they are and how they’re put together. If you just learn the procedures by memorizing a few things, you’ll be hurting later in the class.