Malte Kagan

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As we have seen in examples above, we can represent a function using a graph. Graphs display many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. We typically construct graphs with the input values along the horizontal axis and the output values along the vertical axis.

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. 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). 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. 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. However, the set of all points \left(x,y\right) satisfying y=f\left(x\right) is a curve. The curve shown includes \left(0,2\right) and \left(6,1\right) because the curve passes through those points.

The vertical line test can be used to determine whether a graph represents a function. A vertical line includes all points with a particular x value. The y value of a point where a vertical line intersects a graph represents an output for that input x value. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because that x value has more than one output. A function has only one output value for each input value.

Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. A horizontal line includes all points with a particular y value. The x value of a point where a vertical line intersects a function represents the input for that output y value. If we can draw any horizontal line that intersects a graph more than once, then the graph does not represent a function because that y value has more than one input.

In this text we explore functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use x as the input variable and y=f\left(x\right) as the output variable.

We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown below.

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In cryptography, a scytale is a tool used to perform a transposition cipher, consisting of a cylinder with a strip of parchment wound around it on which is written a message. The ancient Greeks, and the Spartans in particular, are said to have used this cipher to communicate during military campaigns.

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2. Let it drape across your fingers so you can move it easily
3. Complete one full breath (inhale and exhale)
4. Move your fingers to the next bead, breathing in and out once per bead
5. Finish at the guru bead to complete 108 breaths

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