chapter 1: functions & models 1.1 four ways to represent a function
TRANSCRIPT
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.
23456
-10123
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
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”
Dxxfx )(,
Difference Quotient
h
afhaf )()(
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 m3. 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.
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
Example 7• A function f is defined by
• Evaluate f(0), f(1), and f(2) and sketch the graph
1,
1,1)(
2 xx
xxxf
Absolute Value
• |a| = a if a ≥ 0
• |a| = -a if a < 0
• Remember if a is negative, then –a is positive!
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
...
43,11.1
32,87.0
21,63.0
10,39.0
)(
w
w
w
w
wC
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)
Increasing vs Decreasing
• Increasing if f(x1) < f(x2) whenever x1 < x2
• Decreasing if f(x1) > f(x2) whenever x1 < x2