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Special Topics on Functions, Sequences, and Series MAT/117 Version 9
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University of Phoenix Material
Special Topics on Functions, Sequences, and Series The goal of this week is to introduce more concepts and skills about functions that you started learning in the previous week. In addition, you will learn to identify when a list of numbers have an arithmetic or geometric pattern.
Composition of Functions
The composition of functions is denoted with the notation . This operation consists on the idea that the input of one function is another function, or algebraically, this
will be written as . This means “The input of is .” In Cognitive Tutor, you will learn composition of linear functions. However, it is possible to do composition of functions with any type of functions. Here is an example between linear and quadratic functions:
Example
Let and – . So,
( )
Careful: is not the same as . Order matters! See the example below using the same functions above.
( )
One-to-One Functions
Previously, you learned that the basic idea of a function is that “every input has exactly one output,” or
“every x value has exactly one y value.”
For example, in the graph of the function below, you can see every x value corresponds with (matches)
exactly one y value (look closely how each vertical gridline intersects the graph only once; this is called the Vertical Line Test).
Special Topics on Functions, Sequences, and Series MAT/117 Version 9
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If you look closely the graph above, there are different inputs that have the same output (or there are
different x values that correspond to the same y value).
Now, when each input of a function has a different output (or every x value has a different y value), then we say the function has a one-to-one correspondence. Here is an example of a one-to-one function:
In the graph above, every x value has a different y value (look closely how each horizontal gridline intersects the graph only once; this is called the Horizontal Line Test). Understanding one-to-one functions is essential to understand inverse functions, which you will learn in Cognitive Tutor. A real-life example of one-to-one functions is the following: Every person has a unique social security number and each social security number corresponds to a person.
Exponential Functions
When a constant (a fixed number) is raised to a variable (say, x), we are working with exponential
functions. Algebraically, exponential functions have the form , where a > 0 but cannot be 1.
For example, . When the value of is greater than 1, the graph of these functions have this form:
From the graph above, as the x increases, y increases by a factor of a. For such reason, we call the
constant a the growth factor. When the value of a is between 0 and 1 (in other words, a rational number), the graph of these functions have the form
Special Topics on Functions, Sequences, and Series MAT/117 Version 9
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From the graph, as the x increases, y decreases by a factor of a. For such reason, we call the constant a the decay factor. A real-life example of exponential functions that grow by a constant factor is the compound interest you earn from investments. Depreciation of the value of a car is a real-life example of exponential functions that decrease by a constant factor.
Logarithmic Functions
A logarithm represents an exponent. Logarithms evaluate the relevant power that a positive base number is raised to and are defined with the following equation:
Where b is the positive base, x is the number to be evaluated, and y is the relevant power (in other words, exponent). The relationship between these variables could also be written as:
The most often used logarithm is where , so means . This is referred to as
the common logarithm. The base 10 is implied if no subscript number is given in the logarithmic expression.
So, is typically expressed as . The follow charts offers some common logarithms for numbers that are powers of 10:
10 -3
(0.001)
10 -2
(0.01)
10 -1
(0.1)
10 0
(1) 10
1
(10) 10
2
(100) 10
3
(1000)
-3 -2 -1 0 1 2 3
For the logarithmic function , the inverse is
The function has the following graph, shown in the domain [0.1, 6.4].
Special Topics on Functions, Sequences, and Series MAT/117 Version 9
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Another regularly used logarithm is the natural log where the base is , an irrational number. The is typically expressed as . A resource for logarithms from Khan Academy: http://www.khanacademy.org/math/algebra/logarithms/v/introduction-to-logarithms Logarithms have multiple applications in the real world, including seismology, microbiology, nuclear science, and finance.
Sequences and Series
A sequence is a list of numbers with a particular order. Each number in a sequence is called a term and
these are labeled with the notation . Example of an infinite sequence (a function whose domain is the natural numbers):
3, 5, 7, 9, …
This sequence has infinite number of terms where = 3, = 5, = 7, = 9, and so on. If you look closely, the next term of this sequence is 2 more than the previous term. When a sequence is created by adding a constant amount, the sequence is considered an arithmetic sequence. Any term of an arithmetic sequence can be found using the formula:
where is the first term of the sequence, is the position of the term in the sequence, and is the common difference between any two terms. For the sequence above, = 3 and = 2; so,
Do the following to find the 10th term in the sequence infinite 3, 5, 7, 9 …
When a sequence is created by multiplying a constant amount, the sequence is considered a geometric sequence. For example,
2, 4, 8, 16, … If you look closely, the next term of this sequence is 2 times the previous term. Any term of a geometric sequence can be found using the formula:
where is the first term of the sequence, is the position of the term in the sequence, and is the common ratio (division) between any two terms.
For the sequence above, = 2 and = 2 (the ratio between any two terms, such as 4/2); so,
Special Topics on Functions, Sequences, and Series MAT/117 Version 9
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To find the 10th term of the geometric sequence above, do the following:
A series is the sum of the numbers in a sequence. The partial sum of a sequence is denoted with the
notation . For example, to find the sum of the first 4 terms of the infinite sequence 3, 5, 7, 9, …, we write it as
or we can use summation notation and the formula to find each term of the sequence:
∑
Fortunately, there are formulas to quickly find the partial sum of arithmetic and geometric sequences: Partial Sum for an Arithmetic Sequence:
is the first term and is the last term of the sequence. For example, the partial sum of the arithmetic sequence 3, 5, 7, 9 is
Partial Sum for a Geometric Sequence:
is the first term, is the number of terms in the sequence, and is the common ratio. For example, the partial sum of the geometric sequence 2, 4, 8, 16 is
