Review of Functions
Definition of function
Let A and B be two nonempty sets. A function f is a rule that assigns to each element x in A one
and only one element y = f (x) in B.
of the function f .
The set A is called the
The
out the domain.
of f is the set of all possible values of f (x) as x varies through-
CONVENTION: If a function is given by a formula and the domain is not stated explicitly, the domain
is the set of all numbers for which the formula makes sense and defines a real number.
Example 1. Find the domain of the following functions. Write your answer in interval notation.
√
a) f (x) = 2x + 7
x3 + 5x2 − 1
b) f (x) = 2
(x − 3x + 2)2
The letter that represents an arbitrary element in the domain of a function f is called the
. The letter representing an arbitrary element in the range of f is
called the
.
Graph of Functions
When the domain and range of a function are sets of real numbers, the function can be visualized by
its graph.
Definition: The graph of a function f consists of all pairs (x, y) such that x is in the domain of f and
y = f (x).
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Graphs of the Basic Functions:
• Identity Function: y = f (x) = x
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• Quadratic Function: y = f (x) = x2
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• Cubic Function: y = f (x) = x3
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• Square Root Function: y = f (x) =
√
x
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• Rational Function: y = f (x) =
1
x
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• Exponential Function: y = f (x) = ex
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• Logarithmic Function: y = f (x) = ln x
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• Absolute Value Function: y = f (x) = |x|
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The absolute value function is an example of a piecewise defined function:
The Vertical Line Test (VLT): A graph in the xy-plane represents a function of x, if and only if, no
vertical line intersects the curve more than one place.
Example 2. Use the VLT to determine if each of the following graphs is a graph of a function:
a)
b)
A function f is said to be increasing on an interval (a, b) if
It is said to be decreasing on (a, b) if
Concave Up, Concave Down
If the graph of a function bends upward, like the mouth of this face :), we say that the function is
.
If the graph bends downward, like the mouth of this face :(, we say that the function is
Example 3. Use the graph to determine the intervals where the function is increasing and decreasing,
concave up and concave down.
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Transformations of Functions:
• Vertical Shifts
– y = f (x) + c shifts the graph of y = f (x) a distance c units upward.
– y = f (x) − c shifts the graph of y = f (x) a distance c units downward.
• Horizontal Shifts
– y = f (x − c) shifts the graph of y = f (x) a distance c units to the right.
– y = f (x + c) shifts the graph of y = f (x) a distance c units to the left.
• Vertical Stretching
– y = cf (x) stretches the graph of y = f (x) vertically by a factor of c if c > 1.
– y = cf (x) compresses the graph of y = f (x) vertically by a factor of c if 0 < c < 1.
• Reflections
– y = −f (x) reflects the graph of y = f (x) about the x-axis.
– y = f (−x) reflects the graph of y = f (x) about the y-axis.
1√
1 + x by applying the appropriate transformations to the
2
1√
standard function. What is the domain and range of y = 2 −
1 + x?
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Example 4. Sketch the graph of y = 2 −
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Graphers Versus Calculus
We call a graph complete if the portion of the graph that we see in the viewing window of our calculator
suggests all the important features of the graph. Unfortunately, no matter how we may scale the graph,
we can never be certain that some interesting behavior might not be occurring outside the viewing
window or some interesting wiggles aren’t hidden within the curve that we see. Thus, if we use only a
graphing utility, we might overlook important discoveries. This is one reason why we need to carefully
do a mathematical analysis.
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