Activity 2 What is a function? Part 1 The definition of a function A function is a correspondence, or, "rule" between inputs and outputs such that every input corresponds to only one output. This is different from the more general notion of a relation, where an input can be sent to multiple outputs. One can think of a function as a "machine" that accepts inputs and turns them into outputs. For example, consider the function that multiplies every input by 2. If we input into this function, we get our output will be . as the output. If represents a generic number that we input, Function notation It would be cumbersome to have to refer to the function above as "the function that multiplies every input by 2". Therefore, we introduce the idea of function notation as a shorter way to get the same idea across. Remember that we said inputting a generic number x will yield an output of . This relationship tells us exactly what our function does multiplies inputs by 2. If we denote our function by a lowercase letter, say , we can describe what it does as . We would read this as "f of x equals 2x". This is a short way of saying that is the function that, when is the input, the output of the function is . PROBLEM 1 Convert each of the following sentences to function notation. Example. " is the function that divides the input by 3 and adds 1" would be written: a. is the function that subtracts 2 from the input and takes the square root of the result. b. is the function that takes the reciprocal of the input and then takes the absolute value of the result. c. is the function that squares the input, then adds half of the input. . Part 2 Function evaluation One advantage to function notation is that it makes it easy to evaluate functions that is, input numbers or expressions into the function. Example. Suppose we have the function subtracts 1 from the result. . This is the function that squares the input and Let's say we wanted to input 2 into this function. Using the verbal description of the function, we would:2 i. Square the input : 2 squared is 4 ii. Subtract 1 from the result of the last step 4 minus 1 is 3. The output is 3. Using function notation, we can do the following. We are no longer inputting a generic , we are inputting 2. Therefore, we replace all instances of in the functional expression with 2: Inputting variable expressions While we often input numbers into a function, we can also input expressions involving variables. We use the same technique as inputting numbers we replace with the new input. Example. Suppose again that we have the function in place of . We would write: . Or, suppose we want to input Then, we would write Finally, let's input x + h. We would write PROBLEM 2 in place of . and we want to input the variable Suppose . Evaluate the following, expanding algebraic expressions where possible: a. b. c. d. Part 3 Difference quotients Given a function , we can define the difference quotient as: Example. Given the function , its difference quotient is . PROBLEM 3 For each of the functions below, determine the difference quotient (simplify when possible). Show ALL work. a. b. c