Mathematics: Basic Functions

Function and Relation
When expressing functional relationships, the variables x and y are typically used. These values are often written as the coordinates
.
The x-value is the independent variable and the y-value is the dependent variable. A relation is a set of data in which there is not a unique y-value for each x-value in the dataset. This means that there can be two of the same x-values assigned to different y-values.
A relation is simply a relationship between the x and y-values in each coordinate but does not apply to the relationship between the values of x and y in the data set. A function is a relation where one quantity depends on the other.

For example, the amount of money that you make depends on the number of hours that you work. In a function, each x-value in the data set has one unique y-value because the y-value depends on the x-value.

Functions
A function has exactly one value of output variable (dependent variable) for each value of the input variable (independent variable).  The set of all values for the input variable (here assumed to be x) is the domain of the function, and the set of all corresponding values of output variable (here assumed to be y) is the range of the function.  When looking at a graph of an equation, the easiest way to determine if the equation is a function or not is to conduct the vertical line test.  If a vertical line drawn through any value of  crosses the graph in more than one place, the equation is not a function.

Finding the Domain and Range of a Function
The domain of a function 
is the set of all input values for which the function is defined. The range of a function 
is the set of all possible output values of the function—that is, of every possible value of , for any value of x in the function's domain. For a function expressed in a table, every input-output pair is given explicitly. To find the domain, we just list all the x values and to find the range, we just list all the values of . Consider the 

x: -1 | 4 | 2 | 1 | 0 | 3 | 8 | 6
f(x):3 | 0 | 3 | –1 | –1 | | 2 | 4 | 6


In this case, the domain would be {-1, 4, 2, 1, 0, 3, 8, 6}, or, putting them in ascending order, {-1, 0, 1, 2, 3, 4, 6, 8}. (Putting the values in ascending order isn't strictly necessary, but generally makes the set easier to read.) The range would be {3, 0, 3, –1, –1, 2, 4, 6}. Note that some of these values appear more than once. This is entirely permissible for a function; while each value of x must be matched to a unique value of
, the converse is not true. We don't need to list each value more than once, so eliminating duplicates, the range is {3, 0, –1, 2, 4, 6}, or, putting them in ascending order, {–1, 0, 2, 3, 4, 6}.
Note that by definition of a function, no input value can be matched to more than one output value. It is good to double check to make sure that the data given follows this and is therefore actually a function.

Determining a Function
You can determine whether an equation is a function by substituting different values into the equation for x. You can display and organize these numbers in a data table. A data table contains the values for x and y, which you can also list as coordinates. In order for a function to exist, the table cannot contain any repeating x-values that correspond with different y-values.
If each x-coordinate has a unique y-coordinate, the table contains a function. However, there can be repeating y-values that correspond with different x-values. An example of this is when the function contains an exponent. For example, if
,
, and
.

Writing a Function Rule Using a Table
If given a set of data, place the corresponding x and y-values into a table and analyze the relationship between them.
Consider what you can do to each x-value to obtain the corresponding y-value.
Try adding or subtracting different numbers to and from x and then try multiplying or dividing different numbers to and from
.
If none of these operations give you the y-value, try combining the operations. Once you find a rule that works for one pair, make sure to try it with each additional set of ordered pairs in the table. If the same operation or combination of operations satisfies each set of coordinates, then the table contains a function. The rule is then used to write the equation of the function in '
' form.

Direct and Inverse Variations of Variables
Variables that vary directly are those that either both increase at the same rate or both decrease at the same rate. For example, in the functions
are positive, the value of <i>y</i> increases as the value of <i>x</i> increases and decreases as the value of <i>x</i> decreases.<br> Variables that vary inversely are those where one increases while the other decreases. For example, in the functions <br><img src=" />k decreases and decreases as the value of x increases.
In both cases, k is the constant of variation.

Properties of functions
There are many different ways to classify functions based on their structure or behavior. Important features of functions include:
End behavior: the behavior of the function at extreme values ()
· y-intercept: the value of the function at·     

Roots: the values of 
where the function equals zero ()
Extrema: minimum or maximum values of the function or where the function changes direction (

or
)

Classification of Functions

An invertible function is defined as a function,
, for which there is another function,
, such that
. For example, if

the inverse function,
, can be found:

Note that

is a valid function over all values of x.
In a one-to-one function, each value of x has exactly one value for y on the coordinate plane (this is the definition of a function) and each value of y has exactly one value for x.
While the vertical line test will determine if a graph is that of a function, the horizontal line test will determine if a function is a one-to-one function.
If a horizontal line drawn at any value of y intersects the graph in more than one place, the graph is not that of a one-to-one function. Do not make the mistake of using the horizontal line test exclusively in determining if a graph is that of a one-to-one function. A one-to-one function must pass both the vertical line test and the horizontal line test. As such, one-to-one functions are invertible functions.
A many-to-one function is a function whereby the relation is a function, but the inverse of the function is not a function. In other words, each element in the domain is mapped to one and only one element in the range. However, one or more elements in the range may be mapped to the same element in the domain. A graph of a many-to-one function would pass the vertical line test, but not the horizontal line test. This is why many-to-one functions are not invertible.

A monotone function is a function whose graph either constantly increases or constantly decreases. Examples include the functions
,
, or
. A. even function has a graph that is symmetric with respect to the y-axis and satisfies the equation
is any real number and <i>n</i> is a positive even integer. A. odd function has a graph that is symmetric with respect to the origin and satisfies the equation <br><img src=" />a is any real number and n is a positive odd integer.
Constant functions are given by the equation
is a real number. There is no independent variable present in the equation, so the function has a constant value for all <i>x</i>. The graph of a constant function is a horizontal line of slope 0 that is positioned <i>b</i> units from the <i>x</i>-axis. If <i>b</i> is positive, the line is above the <i>x</i>-axis; if <i>b</i> is negative, the line is below the <i>x</i>-axis.<br> Identity functions are identified by the equation <br><img data-cke-saved-src=" />, where every value of the function is equal to its corresponding value of x. The only zero is the point (0, 0). The graph is a line with slope of 1.
In linear functions, the value of the function changes in direct proportion to x. The rate of change, represented by the slope on its graph, is constant throughout. The standard form of a linear equation is
and <i>d</i> are real numbers. As a function, this equation is commonly in the form x to get

, which is the only zero of the function. The domain and range are both the set of all real numbers.
Algebraic functions are those that exclusively use polynomials and roots. These would include polynomial functions, rational functions, square root functions, and all combinations of these functions, such as polynomials as the radicand. These combinations may be joined by addition, subtraction, multiplication, or division, but may not include variables as exponents.

Absolute Value Functions A. absolute value function is in the format
. Like other functions, the domain is the set of all real numbers. However, because absolute value indicates positive numbers, the range is limited to positive real numbers. To find the zero of an absolute value function, set the portion inside the absolute value sign equal to zero and solve for x. An absolute value function is also known as a piecewise function because it must be solved in pieces—one for if the value inside the absolute value sign is positive, and one for if the value is negative. The function can be expressed as

This will allow for an accurate statement of the range. The graph of an example absolute value function,
, is below:

Piecewise Functions
A piecewise function is a function that has different definitions on two or more different intervals. The following, for instance, is one example of a piecewise-defined function:

To graph this function, we'd simply graph each part separately in the appropriate domain. The final graph would look like this:

Note the filled and hollow dots at the discontinuity at
.
This is important to show which side of the graph that point corresponds to.
Because

on the closed interval
,
. The point

is therefore marked with a filled circle, and the point
, which is the endpoint of the rightmost

part of the graph but not actually part of the function, is marked with a hollow dot to indicate this.

Quadratic Functions
A quadratic function is a function in the form
does not equal 0. While a linear function forms a line, a quadratic function forms a parabola, which is a u-shaped figure that either opens upward or downward. A parabola that opens upward is said to be a positive quadratic function and a parabola that opens downward is said to be a negative quadratic function. The shape of a parabola can differ, depending on the values of <i>a</i>, <i>b,</i> and <i>c</i>. All parabolas contain a vertex, which is the highest possible point, the maximum, or the lowest possible point, the minimum. This is the point where the graph begins moving in the opposite direction. A quadratic function can have zero, one, or two solutions, and therefore, zero, one, or two <i>x</i>-intercepts. Recall that the <i>x</i>-intercepts are referred to as the zeros, or roots, of a function.<br> A quadratic function will have only one <i>y</i>-intercept. Understanding the basic components of a quadratic function can give you an idea of the shape of its graph.<br> Example graph of a positive quadratic function, <br><img data-cke-saved-src=" />:


Polynomial Functions
A polynomial function is a function with multiple terms and multiple powers of x, such as:

where n is a non-negative integer that is the highest exponent in the polynomial and
. The domain of a polynomial function is the set of all real numbers. If the greatest exponent in the polynomial is even, the polynomial is said to be of even degree and the range is the set of real numbers that satisfy the function. If the greatest exponent in the polynomial is odd, the polynomial is said to be odd and the range, like the domain, is the set of all real numbers.

Rational Functions
A rational function is a function that can be constructed as a ratio of two polynomial expressions:
, where

and

are both polynomial expressions and
. The domain is the set of all real numbers, except any values for which
. The range is the set of real numbers that satisfies the function when the domain is applied.

When you graph a rational function, you will have vertical asymptotes wherever
and <i>q_n-1</i> are the coefficients of the highest degree terms in their respective polynomials.<br> <br> Square Root Functions<br> A square root function is a function that contains a radical and is in the format <br><img data-cke-saved-src=" />. The domain is the set of all real numbers that yields a positive radicand or a radicand equal to zero. Because square root values are assumed to be positive unless otherwise identified, the range is all real numbers from zero to infinity. To find the zero of a square root function, set the radicand equal to zero and solve for x. The graph of a square root function is always to the right of the zero and always above the x-axis.
Example graph of a square root function,
:

Problems

P1. Martin needs a 20% medicine solution. The pharmacy has a 5% solution and a 30% solution. He needs 50 mL of the solution.
If the pharmacist must mix the two solutions, how many milliliters of 5% solution and 30% solution should be used?
P2. Describe two different strategies for solving the following problem:
Kevin can mow the yard in 4 hours.
Mandy can mow the same yard in 5 hours. If they work together, how long will it take them to mow the yard?
P3. A car, traveling at 65 miles per hour, leaves
Flagstaff and heads east on I-40. Another car, traveling at 75 miles per hour, leaves Flagstaff 2 hours later, from the same starting point and also heads east on I-40. Determine how many hours it will take the second car to catch the first car by:
(a) Using a table.
(b) Using algebra.
 

P1. To solve this problem, a table may be created to represent the variables, percentages, and total amount of solution. Such a table is shown below:

mL solution
% medicine
Total mL medicine
5% solution

0.05


30% solution

0.30

Mixture

0.20

The variable x may be rewritten as
, so the equation

may be written and solved for y. Doing so gives
.
So, 30 mL of 30% solution are needed. Evaluating the expression,

for an
-value of 20, shows that 20 mL of 5% solution are needed.

P2. Two possible strategies both involve the use of rational equations to solve. The first strategy involves representing the fractional part of the yard mowed by each person in one hour and setting this sum equal to the ratio of 1 to the total time needed. The appropriate equation is
, which simplifies as
, and finally as
. So the time it will take them to mow the yard, when working together, is a little more than
2.2 hours.
A second strategy involves representing the time needed for each person as two fractions and setting the sum equal to 1 (representing 1 yard). The appropriate equation is
, which simplifies as
, and finally as
. This strategy also shows the total time to be a little more than 2.2 hours.

P3. (a) One strategy might involve creating a table of values for the number of hours and distances for each car. The table may be examined to find the same distance traveled and the corresponding number of hours taken. Such a table is shown below:



The table shows that after 15 hours, the distance traveled is the same. Thus, the second car catches up with the first car after a distance of 975 miles and 15 hours.

(b) A second strategy might involve setting up and solving an algebraic equation. This situation may be modeled as
. This equation sets the distances traveled by each car equal to one another. Solving for x gives
. Thus, once again, the second car will catch up with the first car after 15 hours.