# Functions

A mathematical relation is a set of ordered pairs . The relation describes which and values can be paired together. The set of all values in the relation is called the domain, and the set of all values in the relation is called the range.

Identify the domain and range of the set of ordered pairs The domain is the set of values, and the range is the set of values, so we have:

Domain = Range = Identify the domain and range of the relation Domain = Range = A function is a special relation. It is a relation for which any particular value is paired only with one particular vallue. That is, a function is a set of ordered pairs for which every value is paired with exactly one value. That is, a function cannot have two ordered pairs with the same -coordinate, but different -coordinates.

Determine if the relation is a function This relation is indeed a function, because every value is paired up with only one value.

Determine if the relation is a function This relation is not a function, as the value of 1 is paired with two different values, 1 and 7

When an equation is a function, it is often expressed in function notation, that is, the output variable is replaced with so that is easily recognized as being a function. is read “ of ” and indicates that the function has been named “ ” and its input variable is . A function need not necessarily always be named . When working with multiple functions, it is common to name them , , and so forth. This helps to keep from using one function when another was intended. An example of a function in function notation: The domain of a function equation is the set of all numbers that can be input into the function, and receive an output. That is, it is any number for which the function is defined. To find the domain of a function equation (generally referred to as simply a function from here on out), it is often easier to find the values of that cannot be input into the function.

Find the domain of the function We can evaluate numbers one by one, but that is certainly not a good method to check an infinite number of possible input values. Instead, we reason that any number can be doubled, so any number can be input into the equation. The domain of this function then, is all real numbers. In symbols, we write this as This means that is an element of , which is the symbol for all “Real numbers”. It is the same as the interval notation Some functions have restrictions that cane be found quickly by understanding their behavior.

Find the domain of the function First, by knowing the properties of the square root, our job is made much easier. We know that we can take the square root of a positive number, and we can take the square root of zero (the square root of 0 is 0). But we cannot take the square root of a negative number. So the radicand (the number, or expression underneath the square root function) must be at least zero. As we learned earlier, this means that we have the inequality: If we solve this inequality, we find our domain!  To write this in interval notation: Find the domain of the function Again, the radicand must be greater than or equal to zero:   So Another function to consider is a fraction with the variable in the denominator. To deal with these functions, we recall that one can not divide by zero, so our domain is restricted and can not include any value that creates a zero in the denominator

Find the domain of the function We can set up our restriction as in inequality using the “not equal to” sign, . Here, the denominator is simply the variable , so we have And there is nothing more to solve. simply can not equal zero. But it can be anything else! This means that we have a union. All of the numbers less than 0 and all of the numbers greater than zero. To write this in interval notation: Find the domain of the function We again will set the denominator not equal to zero: And solve for :   Find the domain of the function Again, we set the denominator not equal to zero, and then solve:    Find the domain of the function In this problem, we have two issues. First, we have a denominator with a variable in it, and second, the denominator has a square root! We can reason our way through the domain restrictions here, noting that the radicand must be greater than or equal to zero. However, the denominator itself can not be zero, so we simply need to remove the equality, and we have our restriction:    To evaluate a function for a particular value, the is replaced with the value and then the expression is simplified. The final result is the corresponding value to the value input. To express that we are evaluating a function for a value, for example 3, we write . This is read “ of 3” or equivalently, “the function evaluated at 3”.

If , find and  Thus, we have the ordered pairs (2,7) and (5,16)

If , find , and   and This gives us the ordered pairs (0,-4), (-2,8) and (1,1)