Functions

represent function

graph, table, and formula

x is

independent variable, domain of a function and it is the input

how to find range

just rearrange the equation and equal with the domain

piece-wise function

Some functions are defined using different equations for different parts of their domain. These types of functions are known as piecewise-defined functions. For example, suppose we want to define a function f with a domain that is the set of all real numbers such that f (x) = 3x + 1 for x ≥ 2 and f (x) = x2 for x < 2. We denote this function by writing f (x) = ⎧ ⎩ ⎨ 3x + 1 x ≥ 2 x2 x < 2 . When evaluating this function for an input x, the equation to use depends on whether x ≥ 2 or x < 2. For example, since 5 > 2, we use the fact that f (x) = 3x + 1 for x ≥ 2 and see that f (5) = 3(5) + 1 = 16. On the other hand, for x = −1, we use the fact that f (x) = x2 for x < 2 and see that f (−1) = 1.

vertical asymptote

just test the denominator

slope of vertical line

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how to find restrictions or holes

you can just plug in the restriction number and solve and the number you are going to find is going to be the restriction.

slope of horizontal line

zero

A function is

A function f consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output. The set of inputs is called the domain of the function. The set of outputs is called the range of the function. Given two sets AA and B,B, a set with elements that are ordered pairs (x,y),(x,y), where xx is an element of AA and yy is an element of B,B, is a relation from AA to B.B. A relation from AA to BB defines a relationship between those two sets. A function is a special type of relation in which each element of the first set is related to exactly one element of the second set. The element of the first set is called the input; the element of the second set is called the output. Functions are used all the time in mathematics to describe relationships between two sets. For any function, when we know the input, the output is determined, so we say that the output is a function of the input. For example, the area of a square is determined by its side length, so we say that the area (the output) is a function of its side length (the input). The velocity of a ball thrown in the air can be described as a function of the amount of time the ball is in the air. The cost of mailing a package is a function of the weight of the package. Since functions have so many uses, it is important to have precise definitions and terminology to study them.

function additional

For a general function f with domain D, we often use x to denote the input and y to denote the output associated with x. When doing so, we refer to x as the independent variable and y as the dependent variable, because it depends on x. Using function notation, we write y = f (x), and we read this equation as "y equals f of x." For the squaring function described earlier, we write f (x) = x2.

function rule

Given an algebraic formula for a function f , the graph of f is the set of points ⎛ ⎝x, f (x)⎞ ⎠, where x is in the domain of f and f (x) is in the range. To graph a function given by a formula, it is helpful to begin by using the formula to create a table of inputs and outputs. If the domain of f consists of an infinite number of values, we cannot list all of them, but because listing some of the inputs and outputs can be very useful, it is often a good way to begin. When creating a table of inputs and outputs, we typically check to determine whether zero is an output. Those values of x where f (x) = 0 are called the zeros of a function. For example, the zeros of f (x) = x2 − 4 are x = ± 2. The zeros determine where the graph of f intersects the x -axis, which gives us more information about the shape of the graph of the function. The graph of a function may never intersect the x-axis, or it may intersect multiple (or even infinitely many) times. Another point of interest is the y -intercept, if it exists. The y -intercept is given by ⎛ ⎝0, f (0)⎞ ⎠. Since a function has exactly one output for each input, the graph of a function can have, at most, one y -intercept. If x = 0 is in the domain of a function f , then f has exactly one y -intercept. If x = 0 is not in the domain of f , then f has no y -intercept. Similarly, for any real number c, if c is in the domain of f , there is exactly one output f (c), and the line x = c intersects the graph of f exactly once. On the other hand, if c is not in the domain of f , f (c) is not defined and the line x = c does not intersect the graph of f . This property is summarized in the vertical line test.

zeros of a function

When creating a table of inputs and outputs, we typically check to determine whether zero is an output. Those values of x where f (x) = 0 are called

y is

dependent variable, range of a function and it is the output related with x