1.1 Four Ways to Represent a Function
A
f is a rule that assigns to each element x in a set D exactly one
element, called f (x), in a set E.
We usually consider functions for which the sets D and E are sets of real numbers.
Definitions Related to Functions:
• The set D is called the
of the function.
• The number f (x) is the
• The
the domain.
and is read
.
of f is the set of all possible values of f (x) as x varies throughout
• A symbol that represents an arbitrary number in the domain of a function f is called an
.
• A symbol that represents a number in the range of f is called a
.
Ways to Visualize a Function:
• A Machine:
• An Arrow Diagram:
• A Graph:
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Example 1. The graph of a function f is shown below.
a) Find the values of f (−2) and f (1).
b) What are the domain and range of f ?
Example 2. Sketch the graph and find the domain and range of each function.
a) f (x) = 3x − 2
b) g(x) = x2 − 2x + 1
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Example 3. If f (x) = x3 and h 6= 0, evaluate
f (a + h) − f (a)
.
h
f (a + h) − f (a)
The expression
is called a
h
quently in calculus.
and occurs fre-
There are four possible ways to represent a function:
• verbally:
• numerically:
• visually:
• algebraically:
Example 4. An airplane takes off from an airport and lands an hour later at another airport. If t
represents the time in minutes since the plane has left the terminal building, let y(t) be the altitude of
the plane. Sketch a possible graph of y(t).
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Example 5. Temperature readings T (in ◦ F) were recorded every two hours from midnight to 2:00PM
in Phoenix on September 10, 2008. The time t was measured in hours from midnight.
t
T
0 2
82 75
4
6 8
74 75 84
10 12
90 93
14
94
a) Use the readings to sketch a rough graph of T as a function of t.
b) Use your graph to estimate the temperature at 9:00AM.
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Example 6. A spherical balloon with radius r inches has volume V (r) = πr3 . Find a function that
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represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r + 1
inches.
Example 7. Express the area of an equilateral triangle as a function of the length of a side.
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Example 8. Find the domain of each function.
√
a) f (x) = x − 5
b) g(x) =
x2
1
−x
Question: Which curves in the xy-plane are graphs of functions?
Answer:
The Vertical Line Test: 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.
Example 9. For each of the following graphs, use the Vertical Line Test to determine if the graph is
the graph of a function or not.
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A
formulas in different parts of their domains.
is a function that is defined by different
Example 10. A function f is defined by
f (x) =
x + 2 x ≤ −1
x2
x > −1
Evaluate f (−2), f (−1), and f (0) and sketch the graph.
of a number a, denoted by
The
real number line. Distances are always positive or 0, so we have
In general, we have
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, is the distance from a to 0 on the
Example 11. Sketch the graph of the absolute value function f (x) = |x|.
Example 12. Find a formula for the function f graphed below.
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A
is a function that jumps from one value to the next. For example,
Symmetry of Functions:
• If a function f satisfies f (−x) = f (x) for every number x in its domain, then f is called an
.
• If a function f satisfies f (−x) = −f (x) for every number x in its domain, then f is called an
.
• The graph of an even function is symmetric with respect to the y-axis.
• The graph of an odd function is symmetric about the origin.
Example 13. Determine whether each of the following functions is even, odd, or neither even nor odd.
a) f (x) = x|x|
b) g(x) =
x2
x4 + 1
c) h(x) = 1 + 3x2 − x5
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The graphs of the functions in the above example are shown below.
A function f is called
It is called
on an interval I if
on I if
Example 14. Given is the graph of the function f . Estimate the interval(s) on which f is increasing.
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