Chapter 1: Functions & Models 1.1 Four Ways to Represent a Function

Function Happens when one quantity depends on another Remember the function machine? Area of a circle is dependent on its radius Human population increases with time Cost of mailing a letter depends on the weight of the letter

Definition A function f is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set E

Functions D and E are sets of real numbers Set D is the domain of the function f(x) is the “value of f at x” and reads “f of x” range of f is set of all values of f(x)

Functions Independent variable – A symbol that represents an arbitrary number in the domain of a function f Dependent variable – A symbol that represents a number in the range of f

Function Machine x f(x) f(x) = …..

Arrow Diagram x a f(x) f(a)

Graph Most common way to visualize a function If f is a function with domain D, then its graph is the set of ordered pairs: **read as “the graph of f consists of all points (x,y) in the coordinate plane such that y=f(x) and x is in the domain of f”

Example 1 (a) Find the values of f(1) and f(5) (b) What are the domain and range of f?

Example 2a Sketch the graph and find the domain and range of

Example 2b Sketch the graph and find the domain and range of

Example 3 IfAnd evaluate

Difference Quotient Represents the average rate of change of f(x) between x = a and x = a+h

Four Ways to Represent a Function 1.verbally (by a description in words) 2.numerically (by a table of values) 3.visually (by a graph) 4.algebraically (by an explicit formula)

Example 4 When you turn on a hot-water faucet, the temperature T of the water depends on how long the water has been running. Draw a rough graph of T as a function of the time t that has elapsed since the faucet was turned on.

Example 5 A rectangular storage container with an open top has a volume of 10 m 3. The length of its base is twice its width. Material for the base costs $10 per square meter; material for the sides costs $6 per square meter. Express the cost of materials as a function of the width of the base.

Example 6a Find the domain of

Example 6b Find the domain of

Functions The graph of a function is a curve in the xy-plane Which curves in the xy-plane are graphs of functions?

Vertical Line Test Used with a graph of a function A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once. Means that for each element in the domain of the function, there is only ONE element in the range

Is it a function?

Piecewise Defined Functions Defined by different formulas in different parts of their domains

Example 7 A function f is defined by Evaluate f(0), f(1), and f(2) and sketch the graph

Absolute Value |a| = aif a ≥ 0 |a| = -aif a < 0 Remember if a is negative, then –a is positive!

Example 8 Sketch the graph of

Example 9 Find a formula for the function 9 (figure 17)

Example 10 In Example C at the beginning of this section in the book, we considered the cost C(w) of mailing a first- class letter with weight w. In effect, this is a piecewise defined function because, from the table of values, we have: Called a step function

Even Function If a function f satisfies f(-x) = f(x) for every number x in its domain, then f is an even function These are symmetric functions with respect to the y-axis

Odd Functions If f satisfies f(-x) = -f(x) for every number x in its domain, then f is called an odd function These are symmetric about the origin (or rotated 180 degrees)

Example 11a Determine if the function is even, odd, or neither

Example 11b Determine if the function is even, odd, or neither

Example 11c Determine if the function is even, odd, or neither

Increasing vs Decreasing Increasing iff(x 1 ) < f(x 2 ) whenever x 1 < x 2 Decreasing iff(x 1 ) > f(x 2 ) whenever x 1 < x 2

Homework P. 20 1, 5-8, 11, 13, odd, 65, 67