Math 142 Business Mathematics II
Minh Kha
Section A8-1.1
Goals: functions and graphs of functions, Increasing, Decreasing, Concavity, Continuity, and Piecewisedefined Functions.
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Functions
Definition 1.1. Definition of function
Let D and R be two nonempty sets. A function f is a rule that assigns to each element x in D one
and only one element y = f (x) in R.
• The set D is called the domain of the function f , which contains all values of the independent
variable (usually x).
The domain D can be thought as the set of inputs too.
• The range of f is the set of all possible values of f (x) as x varies throughout the domain.
Each value f (x) can be considered as an output. Hence, the range of f is the set of all possible
outputs.
Remark 1.2. When the domain of a function is not stated explicitly, the domain is the set of all
numbers x such that the output f (x) makes sense and is a real number.
Example 1. Find the domain of the following functions. Write your answer in interval notation.
√
a) f (x) = x + 9
b) f (x) =
c) f (x) =
x3 + 1
(x − 1)2
x2
x−1
− 5x + 6
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d) f (x) = x−1/5
Some basic functions
• Power function: f (x) = xn
• Polynomial function: f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0
• Rational function: quotient of two polynomial functions, e.g., x2 /(x + 1)
• Absolute value function: f (x) = |x|
• Exponential function: f (x) = ex
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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 2.1. 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).
Graphs of the Basic Functions:
• Identity Function: y = f (x) = x
• Quadratic Function: y = f (x) = x2
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• Cubic Function: y = f (x) = x3
• Square Root Function: y = f (x) =
√
x
• Absolute Value Function: y = f (x) = |x|
The absolute value function |x| is a typical example of a piecewise defined function.
Definition 2.2. Piecewise defined functions are the ones whose domains are divided into two or
more parts with a different rule applied to each part.
Example 2. Another example of piecewise defined function is
(
1/x if x > 0
f (x) =
.
e−x if x ≤ 0.
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.
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Example 3. Use the VLT to determine if the following is a graph of a function:
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Increasing, Decreasing, Concavity, and Continuity
Definition 3.1.
• A function f is said to be increasing on an interval (a, b) (denoted by %) if the graph of the
function rises while moving left to right.
To put it differently, f is increasing on (a, b) if and only if
f (x1 ) < f (x2 ) when a < x1 < x2 < b
• A function f is decreasing on (a, b) (denoted by &) if the graph of the function falls while
moving left to right.
In other words, f is increasing on (a, b) if and only if
f (x1 ) > f (x2 ) when a < x1 < x2 < b
• Concavity
If the graph of a function bends upward, like a smile ^, we say that the function is concave up.
If the graph of a function bends downward, like a frown _, we say that the function is concave
down.
• A function is said to be continuous on an interval if the graph of the function does not have any
breaks, gaps, or holes in that interval. Otherwise, the function is said to be discontinuous.
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Example 4. Use the graph to determine the intervals where the function is increasing and decreasing,
concave up and concave down, continuous.
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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.
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Example 5. a) Graph the functions |x|, −|x|, 2|x|, 2|x| + 1 on the same graph.
b) Graph the functions x2 , (x − 1)2 , (x + 1)2 on the same graph.
c) Graph the functions
√
x,
√
√
x − 1, x − 1 on the same graph.
Example 6. Let g(x) = 2f (x + 1) − 1. Describe which transformations you need to apply to f to obtain
g.
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