Functions

What is a Function? Functions are used extensively in all areas of mathematics At its heart, a function is just a way to match things together based on some rule *In algebra, a function matches numbers, usually based on an equation We can think of the equation as the “rule” that tells us how to match one number to another For example, we discussed equations in two unknowns, such as 3𝑦+ 4𝑥=9; this is one example of a function If we take 𝑥=6, substitute, and solve for 𝑦 we get 𝑦=−5

What is a Function? So, for the rule 3𝑦+4𝑥=9 (“multiply 𝑦 by 3, multiply 𝑥 by 4, then add to get 9”), the number 𝑥=6 is matched to 𝑦=−5 In fact, this is a solution to the above equation and we can represent it as the ordered pair (6,−5) The next slide gives an informal definition of a function

Definition of a Function *A function is a way of matching numbers from a set called the domain of the function, to numbers of another set called the range of the function The matching is carried out such that every number from the domain is matched to exactly one number from the range

Domain and Range of a Function *The domain of a function is all the numbers that we can use to calculate a value (which is a number in the range) *The range of the function is all the numbers that we can expect from calculating with domain values As you study different kinds of functions, it will often be necessary to determine and write the domain and/or the range; these can be expressed in interval notation or in set-builder notation

Domain and Range of a Function Consider the example given earlier: 3𝑦+4𝑥=9 We often use 𝑥 to represent a number from the domain of the function, and 𝑦 to represent a number in the range of the function (though we can use other variables) *Also, functions are often expressed by solving for 𝑦 (or whatever variable is used for the range values) *We can solve the above equation for 𝑦:

Domain and Range of a Function 𝑦=− 4 3 𝑥+3 We may now ask of this function, “what is the domain?” and “what is the range?” Since any two numbers can be multiplied and any two numbers can be added, the domain includes all real numbers *In interval notation we can write that the domain is −∞,∞ , or we can use the notation ℝ, which also means “all real numbers” *Although I won’t show you why this is true at this time, the range of the above function is also −∞,∞ or ℝ

Domain and Range of a Function Now consider this function: 𝑦= 1 𝑥 This function pairs real numbers with their reciprocals *Recall from the Number Properties, however, that one number has no reciprocal; what number was that? *What this means is that the domain of the above function includes all real numbers except zero

Domain and Range of a Function *In interval notation this is −∞,0 ∪ 0,∞ *Or we can use set-builder notation: 𝑥 in ℝ 𝑥≠0 The range also happens to include all real numbers except zero *In set-builder notation: 𝑦 in ℝ 𝑦≠0 Every function has a domain, but the domain may not include all real numbers! Every function has a range, but the range my not include all real numbers!

Graph of a Function Besides an equation such as 𝑦=− 4 3 𝑥+3, functions can also sometimes be represented as a graph *The graph of a function is the set of all matched pairs 𝑥,𝑦 of a function represented by points in the coordinate plane You can think of the graph of a function as a “picture” of the function We will study different kinds of functions, the graph of which is particular to that kind of function For example, the function 𝑦=− 4 3 𝑥+3 is a called a linear function; its graph is a straight line

Graph of a Function

Table of Values A function may also be represented as a table of values This shows selected values from the domain and the corresponding values in the range; the example below shows some values for 𝑦= − 4 3 𝑥+3 𝒙 𝒚=− 𝟒 𝟑 𝒙+𝟑 𝒙,𝒚 −1 𝑦=16/3 (−1,16/3) 𝑦=3 (0,3) 1 𝑦=8/3 (1,8/3) 2 4/3 (2,4/3) 3 (3,−1)

Table of Values A table of values can be used to graph a function by finding some 𝑥,𝑦 pairs and plotting the points in the coordinate plane For many functions, however, we will be able to determine methods that allows us to graph a function without creating a table of values

Function Notation *Finally, we will often use a special notation for functions, called function notation which can be used in place of 𝑦 and has other special uses *The notation looks like this: 𝑓(𝑥) (the 𝑓 isn’t required; other letters can also be used) Think of 𝑓 as the name of the function; the (𝑥) shows us the variable that is used for values from the domain; it is NOT a multiplication! *When you see 𝑓(𝑥) you should say “f of x” The previous linear function can be written using function notation as 𝑓 𝑥 =− 4 3 𝑥+3; remember that this is exactly the same as 𝑦=− 4 3 𝑥+3

Function Notation Suppose that, for 𝑓 𝑥 =− 4 3 𝑥+3, we are required to find the value that corresponds to (or that matches with) 𝑥=6 We can write, “determine 𝑓(6)”, which we read as “determine f of 6” Now, replace 𝑥 in the equation by 6: 𝑓 6 =− 4 3 ⋅6+3 If we now evaluate, we get 𝑓 6 =−5; a solution is (6,−5) You will see later that we can replace 𝑥 by anything that can represent a number, even other variable expressions

Relations Relations are pairs of numbers that are related in some way A function is a special kind of relation This means that, like a function, number pairs come from a set called the domain and from a set called the range Also like a function, they can be represented as ordered pairs, 𝑥,𝑦 , where 𝑥 is a number from the domain and 𝑦 is a number from the range

Guided Practice Identify the domain and range of the given relation as sets of numbers. 3,1 , 4,−4 , 3,0 , (2,0) Domain: Range: 4,−1 , −3,−2 , 1,1 , 1,−2 Domain: Range:

Guided Practice Determine whether the given relation is a function. 𝒙 3 1 4 2 𝑦 −3 𝒙 −4 3 9 4 𝑦 1 −2

Concentrate!