Section 1.1 Functions and Change

Definition of function A function is a rule that assigns a unique output to each input We say that the output is a function of the input The inputs and outputs are often represented by variables The set of all inputs is called the domain and the set of all outputs is called the range A function can be described using words, data in a table, points on a graph, or a formula

What are the domain and range of the above function? Month 1 2 3 4 5 6 Gas $2.13 $2.16 $2.26 $2.29 $2.41 $2.48 What are the domain and range of the above function? How can we describe month 3 in function notation?

Behavior of Functions Increasing Decreasing Constant Concave Up Concave Down Examples: y = x, y = x3, y = ex

Linear Functions Has a constant rate of change Slope y = f(x) = mx + b where m is the slope and b is the vertical intercept Now And is called the difference quotient

Delta Notation Δ=“change in” Slope Delta Notation Δ=“change in” Difference Quotient

Example Suppose we know the population of a city is 23,000 in 1982 and 21,000 in 1986 Assuming the population has been declining at a constant rate since 1970, find a formula for the population as a function of time, t Let t be the number of years since 1970 (i.e. t = 0 corresponds to the year 1970) Use your model to determine the population in the year 2000 When will the population be 0?