Mathematics: Advanced Functions

Step Functions
The double brackets indicate a step function. For a step function, the value inside the double brackets is rounded down to the nearest integer.

The graph of the function

appears as shown on the left below. In comparison

is on the right below. The coefficient of 2 shows that it's stretched vertically by a factor of 2 (so there's a vertical distance of 2 units between successive
'steps'). The coefficient of

in front of the x shows that it's stretched horizontally by a factor of 3 (so each
'step' is three units long), and the

shows that it's displaced one unit to the right.

Transcendental functions
Transcendental functions are all functions that are non-algebraic. Any function that includes logarithms, trigonometric functions, variables as exponents, or any combination that includes any of these is not algebraic in nature, even if the function includes polynomials or roots.

Exponential Functions
Exponential functions are equations that have the format
, where base

and
.

The exponential function can also be written
. Recall the properties of exponents, like the product of terms with the same base is equal to the base raised to the sum of the exponents:

and a term with an exponent that is raised to an exponent is equal to the base of the original term raised to the product of the exponents:
. The graph of an example exponential function,
, is below:

Note in the graph that the y value approaches zero to the left and infinity to the right.

One of the key features of an exponential function is that there will be one end that goes off to infinity and another that asymptotically approaches a lower bound.

Common forms of exponential functions include:

Geometric sequences:
, where

is the value of the nth term,

is the initial value, r is the common ratio, and n is the number of terms. Note that
.

Population growth:
is the initial population, <i>e</i> is the mathematical constant known as Euler" />r is the growth rate.

Compound interest:
, where

is the account value at a certain number of time periods
, P is the initial principle balance, r is the interest rate, and n is the number of times the interest is applied per time period.
General exponential growth or decay:
, where

is the future count, a is the current or initial count, r is the growth or decay rate, and t is the time.

For example, suppose the initial population of a town was 1,200 people. The population growth is 5%. The current population is 2,400. To find out how much time has passed since the town was founded, we can use the following function:
.

The general form for population growth may be represented as
, where

represents the current population, a represents the initial population, r represents the growth rate, and t represents the time. Thus, substituting the initial population, current population, and rate into this form gives the equation above.

The number of years that have passed were found by first dividing both sides of the equation by 1,200. Doing so gives
. Taking the natural logarithm of both sides gives
. Applying the power property of logarithms, the equation may be rewritten as
, which simplifies as
. Dividing both sides of this equation by 0.05 gives
. Thus, approximately 13.86 years passed.


Logarithmic Functions
Logarithmic functions are equations that have the format
may be any number except one; however, the most common bases for logarithms are base 10 and base <i>e</i>. The log base <i>e</i> is the natural logarithm, or <i>ln</i>, expressed by the function <br><img data-cke-saved-src=" />.
Any logarithm that does not have an assigned value of b is assumed to be base 10:
. Exponential functions and logarithmic functions are related in that one is the inverse of the other. If
, then
. This can perhaps be expressed more clearly by the two equations:

and
.

The following properties apply to logarithmic expressions:




Trigonometric Functions
Trigonometric functions are periodic, meaning that they repeat the same form over and over. The basic trigonometric functions are sine (abbreviated ‘sin'), cosine (abbreviated ‘cos'), and tangent (abbreviated ‘tan'). The simplest way to think of them is as describing the ratio of the side lengths of a right triangle in relation to the angles of the triangle.

Using sine as an example, trigonometric functions take the form
amplitude is simply equal to <i>A</i>. The period is the distance between successive peaks or troughs, essentially the length of the repeated pattern. In this form, the period is equal to <br><img src=" />C, this is the phase shift or the horizontal shift of the function. The last term, D, is the vertical shift and determines the midline as
.

For instance, consider the function
. Here
,
,
, and
, so the midline is at
, the amplitude is
, and the period is
. To graph this function, we center the sine wave on the midline and extend it to a height above and below the midline equal to the amplitude—so this graph would have a minimum value of

and a maximum of
.

So, the function would be graphed as follows:


Manipulation of Functions
Translation occurs when values are added to or subtracted from the x or y values. If a constant is added to the y portion of each point, the graph shifts up. If a constant is subtracted from the y portion of each point, the graph shifts down. This is represented by the expression
is a constant. If a constant is added to the x portion of each point, the graph shifts left. If a constant is subtracted from the x portion of each point, the graph shifts right. This is represented by the expression <br><img data-cke-saved-src=" />, where k is a constant.

Stretching, compression, and reflection occur when different parts of a function are multiplied by different groups of constants. If the function as a whole is multiplied by a real number constant greater than 1, (), the graph is stretched vertically. If k in the previous equation is greater than zero but less than 1, the graph is compressed vertically. If k is less than zero, the graph is reflected about the x-axis, in addition to being either stretched or compressed vertically if k is less than or greater than -1, respectively. If instead, just the x-term is multiplied by a constant greater than 1 (), the graph is compressed horizontally. If k in the previous equation is greater than zero but less than 1, the graph is stretched horizontally. If k is less than zero, the graph is reflected about the y-axis, in addition to being either stretched or compressed horizontally if k is greater than or less than -1, respectively.

Algebraic Theorems
According to the fundamental theorem of algebra, every non-constant, single variable polynomial has exactly as many roots as the polynomial's highest exponent. For example, if  is the largest exponent of a term, the polynomial will have exactly 4 roots. However, some of these roots may have multiplicity or be non-real numbers.

For instance, in the polynomial function
, the only real roots are 1 and -1. The root 1 has multiplicity of 2 and there is one non-real root (
).

The remainder theorem is useful for determining the remainder when a polynomial is divided by a binomial. The remainder theorem states that if a polynomial function
is a real number, the remainder of the division will be the value of <i>f</i>(<i>a</i>).<br> If <br><img src=" />a is a root of the polynomial.

The factor theorem is related to the remainder theorem and states that if

then (
) is a factor of the function.

According to the rational root theorem, any rational root of a polynomial function
with integer coefficients will, when reduced to its lowest terms, be a positive or negative fraction such that the numerator is a factor of

and the denominator is a factor of
. For instance, if the polynomial function

has any rational roots, the numerators of those roots can only be factors of 4 (1, 2, 4), and the denominators can only be factors of 1 (1). The function in this example has roots of 1

and -2
.

Applying the Basic Operations to Functions
For each of the basic operations, we will use these functions as examples:

and
.

To find the sum of two functions f and g, assuming the domains are compatible, simply add the two functions together:

To find the difference of two functions f and g, assuming the domains are compatible, simply subtract the second function from the first:
.

To find the product of two functions f and g, assuming the domains are compatible, multiply the two functions together:
.

To find the quotient of two functions f and g, assuming the domains are compatible, divide the first function by the second:
.

The example given in each case is fairly simple, but on a given problem, if you are looking only for the value of the sum, difference, product or quotient of two functions at a particular x-value, it may be simpler to solve the functions individually and then perform the given operation using those values.

The composite of two functions f and g, written as

simply means that the output of the second function is used as the input of the first. This can also be written as
. In general, this can be solved by substituting

for all instances of x in

and simplifying. Using the example functions

and
, we can find that

or

is equal to
, which simplifies to
.
It is important to note that

is not necessarily the same as
. The process is not always commutative like addition or multiplication expressions. It can be commutative, but most often this is not the case.

P1. A professor wishes to invest $20,000 in a CD that compounds annually. The interest rate at his bank is 1.9%. How many years will it take for his account to reach $50,000?
P2. Suppose a new strain of bacteria, after  days, shows a growth rate of 10%. The current count for the new bacteria strain is 100. How many days will pass before the count reaches 1 million bacteria?

P3. Each of the following functions cross the x- and y-axes at the same points.
Identify the most likely function type of each graph
(a)

(b)

(c)

(d)

P4. Given the functions
,
, and
, perform the following operations and write out the resulting function:
(a) Shift

4 units to the left and 1 unit up, then compress the new function by a factor of 1/2
(b)

(c)

(d)


P1. In order to solve this problem, the compound interest formula should be evaluated for a future value of $50,000, principal of $20,000, rate of 0.019, and number of years of t. The exponential equation may then be solved by taking the logarithm of both sides. The process is shown

Dividing both sides of the equation by 20,000 gives
. Taking the logarithm of both sides gives
. Dividing both sides of this equation by

gives
.
Thus, after approximately 49 years, the professor's account will reach $50,000.
P2. The problem may be solved by writing and solving an exponential growth function, in the form,
, where

represents the future count, a represents the current count, r represents the growth rate, and x represents the time. Once the function is evaluated for a future count of 1,000,000, a current count of 100, and a growth rate of 0.10, the exponential equation may be solved by taking the logarithm of both sides.
The problem may be modeled with the equation
. Dividing both sides of the equation by 100 gives
. Taking the logarithm of both sides gives
. Dividing both sides of this equation by log(1.10) gives
.
Thus, after approximately 97 days, the bacteria count will reach 1 million.

P3. (a) Exponential function – positive, increasing slope
Linear function – positive, continuous slope.
Polynomial function (odd degree) – positive, changing slope. Note that the graph goes off to infinity in opposite quadrants I and III, thus it is an odd degree.

(d) Logarithmic function – positive, decreasing slope

P4. (a) Shifting 
to the left 4 units is the same as

and shifting the function up one unit is
. Combining these and multiplying by ½ results in the following:

Factor , noting that it is a perfect square, and be sure to note the constraint on 
due to the original denominator of the rational expression:

Evaluate the composition as follows:





(d) Note the constraint on  due to the original denominator of the rational expression: