The graphs and sample table values are included with each function shown below. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. For these definitions we will use x as the input variable and y=f\left(x\right) as the output variable. Some of these functions are programmed to individual buttons on many calculators. In your example, there are only two rows with a t and your expression will have two terms: ( X Y ¯ Z) ( X ¯ Y ¯ Z) Now you have a logical formula for your truth table. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Pick out the rows where a t appears in the rightmost column, and write down a disjunctive normal form. When working with functions, it is similarly helpful to have a base set of building-block elements. When learning to do arithmetic, we start with numbers. When learning to read, we start with the alphabet. In this text we explore functions-the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. Any horizontal line will intersect a diagonal line at most once. The curve shown includes \left(0,2\right) and \left(6,1\right) because the curve passes through those points. However, the set of all points \left(x,y\right) satisfying y=f\left(x\right) is a curve. For example, the black dots on the graph in the graph below tell us that f\left(0\right)=2 and f\left(6\right)=1. If the function is defined for only a few input values, then the graph of the function is only a few points, where the x-coordinate of each point is an input value and the y-coordinate of each point is the corresponding output value. The graph of the function is the set of all points \left(x,y\right) in the plane that satisfies the equation y=f\left(x\right). The most common graphs name the input value x and the output value y, and we say y is a function of x, or y=f\left(x\right) when the function is named f. We typically construct graphs with the input values along the horizontal axis and the output values along the vertical axis. The visual information they provide often makes relationships easier to understand. Graphs display many input-output pairs in a small space. Identify the graphs of the toolkit functionsĪs we have seen in examples above, we can represent a function using a graph.Verify a one-to-one function with the horizontal line test.Verify a function using the vertical line test.
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