In this case, no matter what input x we fit into the rule, we just output the same constant k over and over again. However, if the slope is equal to zero then the rule simplifies and y is just the constant k. This becomes transparent in a moment when we discuss the graph of the function. 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. The range of the function by contrast is the set of all outputs y produced by applying the rule. For example, for linear function, the rule makes sense for all real numbers x. The domain of f is the set or collection of all real numbers x that are valid inputs for the rule f of x. We now introduce the concepts of domain and range of a function. So we get one minus five minus 35 for g minus two-fifths, one and eight for h. You can try this for g and h with those same inputs zero, one and six for x. f of six means input six and your output 15. f of one means input one for x and you output five. 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. It just changes the way we describe it algebraically. 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. We could choose different constants to get a different function say g and another choice of constants to get another function say h. For example, we could take m equals two and k equals three and then f of x is 2x plus three. 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. The function takes an input x, multiplies it by m and then adds k. For example, consider the following rule for f where m and k are constants. 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. 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. 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 feed in an input, say x, and set the process in motion. If it helps, you can think of the function f as a factory for processing or crunching numbers. 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. So let's get started with the fundamental idea of a function. You'll quickly become expert at creating or sketching graphs of functions even just in your mind's eye. 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. Only certain values might be created leading to the notion of the range. 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. 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.
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