In this video, we'll introduce and discuss functions which are formal mathematical devices for capturing precisely, the relationships that occur or might occur especially if you're making predictions, between quantities or measurements. Functions typically involve formulae built up in a variety of ways and my only makes sense for certain numbers leading to the notion of the domain. Only certain values might be created leading to the notion of the range. The Cartesian plane helps us visualize relationships, using what we call formally the graph of a function which provides a powerful pictorial representation that gives great insight. You'll quickly become expert at creating or sketching graphs of functions even just in your mind's eye. So let's get started with the fundamental idea of a function. A function f is a rule or process that takes an input, typically a real number x, and produces an output also typically a real number denoted by f of x. If it helps, you can think of the function f as a factory for processing or crunching numbers. We feed in an input, say x, and set the process in motion. The function f transforms x into the output f of x and we can do this as many times as we like with as many different numbers x as we like, processing them and churning out lots of outputs. We have an arrow notation with x and an arrow with a little stick at the left-handed end pointing to f of x and we say f takes x to f of x and it's typical to call the output y. We also call the input x, the independent variable and the output y, the dependent variable and it's common to write y equals f of x so that the independent and dependent variables are linked by an equation. For example, consider the following rule for f where m and k are constants. The function takes an input x, multiplies it by m and then adds k. We call this a linear function because when you plot y equals mx plus k in the x y plane, you get a line with slope m and y intercept k. For example, we could take m equals two and k equals three and then f of x is 2x plus three. We could choose different constants to get a different function say g and another choice of constants to get another function say h. Notice, we can rewrite the rule for h as seven x minus two over five if we want, which doesn't change the value produced by the function. It just changes the way we describe it algebraically. Here's some practice evaluating the rules of these functions for example, f of zero means input 0 for x and you output two times zero plus three equals three. f of one means input one for x and you output five. f of six means input six and your output 15. You can try this for g and h with those same inputs zero, one and six for x. So we get one minus five minus 35 for g minus two-fifths, one and eight for h. We now introduce the concepts of domain and range of a function. The domain of f is the set or collection of all real numbers x that are valid inputs for the rule f of x. For example, for linear function, the rule makes sense for all real numbers x. So the domain is the whole real line. The range of the function by contrast is the set of all outputs y produced by applying the rule. For example, for a linear function, provided the slope m is non-zero, it's easy to see that the range is again all the real numbers. This becomes transparent in a moment when we discuss the graph of the function. However, if the slope is equal to zero then the rule simplifies and y is just the constant k. In this case, no matter what input x we fit into the rule, we just output the same constant k over and over again. So the range is the set consisting of one real number k. We also need to discuss graphs of functions. The graph of the function f is the set of all points x y in the Cartesian plane such that y equals f of x. The graph captures visually the relationship between the input x and the output y. For example, the graph of the linear function is, as you expect, a straight line. If the slope m is non-zero then the line can slope upwards or downwards and you can see from the graph that all points on the vertical y-axis become outputs for the function. That's why we observed earlier that for non-zero slope, the range becomes the whole real line. However, if the slope is zero then the graph is a horizontal line passing through the y-axis at some constant k and this is the only value produced by the function which is why we observed earlier that the range is the set consisting of k in this case. Let's look at an example of a quadratic function where the rule is a quadratic expression involving the input x. Take for example, f of x equals x squared minus 4x plus three. If we look at some values of various inputs, we see that f of zero is three and f of one is zero and you can work out some other values. f of two is minus one, f of three is zero, f of minus one is eight and f of minus two is 15. Now the rule for this function, indeed any quadratic function, makes sense again for all real inputs x so the domain is all the real line. But what's the range? The answer is not at all obvious. Let's try rewriting the rule in a different form by completing the square. The coefficient of x is minus four and half of minus four is minus two, so we add and subtract minus two squared. Just before the plus three, without changing the overall value of the left-handed side, we then combine the plus four with the first two terms to get x squared minus 4x plus four which is a perfect square and the minus four with a plus three at the end and the left-handed side becomes x minus two all squared minus one. And so we're able to rewrite f of x in a very useful form. But x minus two all squared is always greater than or equal to zero, that's true for the square of any number. So f of x which is one less is always greater than or equal to minus one. It follows that the range must be the interval of all real numbers greater than or equal to minus one expressed here using interval notation and this will be confirmed shortly by drawing the graph of f. Note, from a different point of view, we can simply factorize f of x as x minus three times x minus one. So the graph must be a parabola that crosses the x axis, that is when y is equal to zero, at x equals three and x equals one. We can plot some points and join them up to get a smooth parabola. Note that the apex, the lowest point, occurs when x is equal to two in order to minimize y equals x minus two squared minus one in fact, with minimum value minus one. The domain is all of r and the range is the interval from minus one to infinity. There's another important family of curves known as hyperbolas. The simplest example is the graph of the reciprocal function y equals f of x equal to one over x. So f of one is one over one which is just one, f of two is just one half, f of three is a third and so on. Notice that if you reciprocate the reciprocal, you get back to where you started. So f of a half which is one over half is two, f of a third is three and so on. f of minus one is just minus one, f of minus two is minus a half and f of minus a half is minus two and so on. But what about f of zero? In fact, it's undefined because you're not allowed to divide by zero. x equals zero is not allowed as an input. This is the only restriction so the domain of f is the set of all non-zero real numbers expressed this way using set notation and also this way using what's called slash notation, where whatever is being slashed is removed from the set. So in this case, we remove zero from the real line and can express the domain also as the union of two intervals, putting together the positive and negative reals avoiding zero. What does the reciprocal function produce as outputs? Well, just all non-zero real numbers. So the range in fact is the same as the domain in this example. Thinking about the graph, we can plot a few points x y where y is the reciprocal of x and then join the points with a smooth curve which in fact becomes two disconnected pieces called the branches of the hyperbola. One branch where everything is positive and another branch where everything is negative. Notice that the branches never touch the x and y-axis. The branches get closer and closer to the axis as you move further and further away from the origin. The axes are called asymptotes. The x-axis is a horizontal asymptote and the y-axis is the vertical asymptote. We've covered a lot of ground in just a few minutes. We have discussed functions which are rules for producing outputs from inputs, the domain, the set of permissible inputs, the range, the set of outputs produced and the graph of the function which is a set of points x y in the Cartesian plane where x is the input and y is the output. We illustrated these ideas using linear functions whose graphs are lines, a quadratic function whose graph is a parabola and the reciprocal function whose graph is a hyperbola, forming two branches with the x and y axis acting as asymptotes. There are more details in the notes, please read and digest them and when you're ready please attempt the exercises. Thank you very much for watching and I look forward to seeing you again soon.
