Chapter 1.1

Section 1.1 serves to introduce and define functions and related concepts. The mathematical skills involved include the following: identifying functions, identifying functions from equations, evaluating functions, determining the domain of a function, and an introduction to difference quotient.  

Some basic operational definitions:

Relation: A correspondence between two quantities or properties that can be predicted with a rule.

Function: A relation between two sets of numbers, the domain and range, in which for each domain value only a single corresponding range value may exist. The values of the domain and range are called inputs and outputs, respectively. Input and output may be represented with an ordered pair (x,y) in which x is an input and y is an output. Functions may be represented verbally (a sentence,) in a table/list of ordered pairs, with points on a graph, or with an equation using two variables. The following are defining characteristics of functions:

• For each possible domain input value (see below implied domain) a corresponding range output value must exist
• For each input value, only one corresponding output value may exist (the converse is not true.)

Function Notation: A method of naming equations that represent functions. For example, consider the equation y = x + 2 which describes y as a function of x. The rule "x+2" constitutes the function's identity, and it is modifying the input/independent value. Therefore you can rewrite the equation as f(x) = x + 2, where f is simply the name of the function, and f(x) is the value produced after a value for x is processed by the rule, x+2. From this, it can be said that f(x) = y.

Implied Domain: When the domain of a function is not explicitly stated, it can be implied by the expression that defines the function. In many cases this may result in excluded values that cannot be used as inputs in the function being considered.

Identifying Functions

When identifying functions, regardless of the way the (potential) function's data is displayed, you should begin by making sure it meets the above stated definition, especially the two bulleted points.

Example I: The input value x represents the average daily temperature in the month of December for a year, and y represents the total inches of snowfall for that month/year.

Answer: Not a function - note that while average daily temperature may be the same two years in a row, this does not necessarily guarantee the exact same amount of snowfall. Therefore, each input value does not correspond to a single, specific output value.

Example II: Consider the relationship depicted by the graph below:

Answer: This is a function - for each visible horizontal input value for x, a single corresponding vertical y value is depicted. A good way to verify this is by considering imaginary vertical lines at each visible x value. If at any input the relation line crosses more than once, you know it is not a function because multiple outputs are corresponding to a single input.

Identifying Functions from Equations

When considering whether a mathematical equation represents a function, the process can be somewhat more involved, though it operates by essentially the same logic. Once again you must determine whether each possible input corresponds to a single output. The two methods for doing so are either to graph the equation and employ the vertical line test, or to analyze the equation. However, both methods require the equation to be in function notation form (or "y =" form.) Generally this simply means you must solve for/isolate y.

Example I: Consider the following equation and determine whether it represents a function: 

  

At this point you can either graph the equation using a calculator, or more simply just analyze the equation's components to determine whether or not it is a function. 

Answer: This is not a function. If you recognized the original equation as that of a circle from the beginning, then you would have already known it would fail the vertical line test. However, the plus or minus symbol would be another indication, because it implies that no matter what input is chosen, both a positive and a negative output would result.

From this example, we can draw some conclusions concerning equations and functions:

• If one side of the equation can be simplified to a single term with term y squared or if a plus/minus sign exists on either side of the equation, it is not a function
• If y is ever within an absolute value symbol, the equation is not a function because it implies either a positive or negative output value is possible

Example II: However, further consider the following example:

Now we have several exponents/roots affecting our equation, including an initial y-cubed term. Once again, you may at this point either graph or analyze. For the purpose of demonstration, here is the graph of the above equation:

As you can see, the equation does indeed represent a function. While graphing the equation did work, analysis would have been quicker. Consider a given input for x. Regardless of its initial sign, the input would be squared and become positive, then negative again due to the sign. Finally, the cube root would be applied, getting a (single) negative answer. Recall that odd numbered roots/exponents retain the sign of the value being modified, while even exponents do so only for positive values, and even roots produce two answers.

Evaluating Functions

To evaluate a function for an input value is simply to apply that function's "rule" to the input. This is frequently requested via function notation, where f(x)={expression modifying x} and you are given f(input value). The only real step here is computing the expression with the input substituted in for x.

Example I:

Here you are told to find the function (f) of 5 based on the "rule" ln(x). This is a simple one-step problem requiring a calculator due to the natural logarithm. 

The input you are told to evaluate may not always be a simple value as shown above. In some cases, variables, expressions, or even other functions may instead be evaluated. In this case, a numerical answer may be unnecessary. An example of this is shown below:

Example II: The below example demonstrates the evaluation of an expression for a given function.

As you can see, the answer is still an expression but has been successfully evaluated in terms of the original quadratic function. The skills involved in this process are fundamental skills of algebra.

Determining the Domain of a Function

As defined above, the implied domain is the set of all (usually real) numbers that may be used in the equation of a function as input values. That is to say, the function of these numbers will equal the range of the function. The domain may also be explicitly stated. There are two primary rules to be followed when determining implied domain from a function's equation:

• values that result in the even root of a negative number (ex. 4th Root[-1])
• values that result in division by zero

A useful way of representing domain is using set-builder notation, as shown below:

The above notation is read as "the set of all x values in the real numbers such that x is greater than or equal to 3." The conditions may be modified by the format is generally an applicable shorthand.

Example I:

Answer: {x | x is not = 5} When dividing by a variable or variable expression, you should always consider cases where the input will result in dividing by zero; a mathematical impossibility. Such input values are implied to be excluded from the function's domain.


Example II:

Answer: { x | x ≥ 1} At the first sign of a variable under an even radical symbol, you should begin checking for values of x that would result in the even root of a negative, and then exclude those values from the implied domain.

One last important detail of domain is that in some cases, such a functions involving units, the domain may be implied by the context. For example, in equations employing units of time or volume, no inputs may be used that result in negative values.

Introduction to Difference Quotient

Difference quotient is a concept that will become fundamental in the early studies of Calculus, and is used in the concept of derivatives. It can in some ways be thought of as an interval-based slope calculation. For example, it could be used on a displacement vs. time graph to determine instantaneous velocity. However, for our purposes we will only be evaluating functions through its form, which contains two function evaluations in one expression:

  To explain where h is coming from, refer to the graph depicted below:

Example 1: For the following function, evaluate a difference quotient:

As shown above, it is simply evaluating a function twice and using the results in an expression; a relatively straightforward process.