Math 142: Lecture 1-Functions
Yanfang Yang
June 2nd, 2015
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function and graph
Definition 1.1. Let D and R be two nonempty sets. A function f from D to R 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 f .
• f (x) is called the value of f at x.
• The set D can be thought as a set of inputs. Then the values f (x) are outputs.
• The set of all possible outputs is called the range of f
Example 1.1. The function f (x) = 2x2 + 5 says that
• take x
• multiply it by itself
• multiply the result by 2; and
• add 5 to the result.
Some Basic Functions:
• Linear Function: f (x) = x
• Quadratic Function:f (x) = x2
• Polynomial Function: f (x) = an x2 + an−1 xn−1 + · · · + a1 x + a0
• Square Root Function: f (x) =
• Cube Root Function: f (x) =
• Rational Function: f (x) =
√
√
3
x
x
x
x+1
• Absolute Value Function: f (x) = |x|
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• Exponentia Function: f (x) = rx
• Power function: f (x) = xr
Note that there is a substantial difference between the exponential function rx and the power function
xr
A symbol that represents an arbitrary number in the domain of a function f is called an independent
variable. A symbol that represents a number in the range of f is called a dependent variable.
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 gives a real number
Example 1.2. Find the domain of the following functions. Write your answer in interval notation.
• f (x) =
• f (x) =
• f (x) =
x2 −4x+4
x2 −4
√
15 − 7x.
x
4x2 −28x−40
A method for visualizing a function is the graph of a function, defined by
Definition 1.2. The graph of a function f consists of all (x,y) such that x is in the domain of f and y
= f (x).
Example 1.3. Graphs of some basic functions.
• f (x) = x;
2
• f (x) = x2 ;
• f (x) =
√
x.
Proposition 1.1. 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 1.4. Use the VLT to determine if the graph is a graph of a function.
40
30
20
10
0
−10
−20
−30
−40
0
2
1
2
3
4
5
6
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increasing, decreasing, concavity, and continuity
Definition 2.1. A function y = f (x) is called increasing (denoted by %) on the interval (a, b) if
the graph of the function rises while moving left to the right or, equivalently, if f (x1 ) ≤ f (x2 ) when
a < x1 < x2 < b.
A function y = f (x) is called decreasing (denoted by nearrow) on the interval (a, b) if the graph of the
function falls while moving left to the right or, equivalently, if f (x1 ) ≥ f (x2 ) when a < x1 < x2 < b.
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If the graph of a function bends upward (like a happy face ^), we say that the function is concave
up.
If the graph bends downward (like a sad face _), we say that the function is concave down.
Definition 2.2. Piecewise-Defined Functions: Functions whose domains are divided into two or
more parts with a dierent rule applied to each part. To graph, graph each rule over the appropriate
portion of the domain.
Example 2.1. An example of a piecewise-defined function is the absolute value function f(x) = —x—
which is dened by
f (x) =

 −x, x < 0
 x, x ≥ 0.
Example 2.2. Use the graph to determine the intervals where the function is increasing and decreasing,
concave up and concave down.
1
0.5
0
−0.5
−1
−1.5
−2
0
3
0.2
0.4
0.6
0.8
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Transformations of Functions
Vertical and Horizontal Shifts: Suppose c > 0. To obtain the graph of
• y = f (x) + c, shift the graph of y = f (x) a distance c units upward.
• y = f (x) − c, shift the graph of y = f (x) a distance c units downward.
• y = f (x − c), shift the graph of y = f (x) a distance c units to the right.
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• y = f (x + c), shift the graph of y = f (x) a distance c units to the left.
Expansion, Contraction and Reflecting: Suppose c > 1. To obtain the graph of
• y = cf (x), stretch the graph of y = f (x) vertically by a factor of c.
• y = 1c f (x), shrink the graph of y = f (x) vertically by a factor of c.
• y = −f (x), reflect the graph of y = f (x) x-axis.
Example 3.1.
• Graph y =
√
x, y =
√
x + 1, and y =
√
x − 2 on the same graph.
• Graph y = x2 , y = (x − 2)2 , and y = (x + 12 )2 on the same graph.
• Graph y = |x|, y = 2|x|, and y = −2|x| on the same graph.
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• if f (x) =
√
√
x and g(x) = 3 x − 2 + 3, how was f (x) transformed to get g(x).
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