Functions in Precalculus

Introduction to Functions

Functions are a foundational concept in precalculus, describing relationships between variables where each input is associated with exactly one output. Understanding functions is essential for analyzing mathematical models and real-world phenomena.

• Functions all around us: Functions can model height as a function of age, temperature as a function of date, and postage as a function of weight.

• Definition of Function: A function is a rule that assigns to each element x in a set A exactly one element, called f(x), in a set B.

• Evaluating a Function: To evaluate a function, substitute the input value into the function's rule.

• The Domain of a Function: The domain is the set of all possible input values for which the function is defined.

Definition and Notation

A function is a rule that assigns to each element x in a set A exactly one element, called f(x), in a set B.

• The symbol f(x) is read as "f of x" and is called the value of f at x.

• The set A, containing the elements x, is called the domain of the function.

• The set of all possible outputs is called the range of the function.

• Independent variable: x

• Dependent variable: f(x)

Machine Diagram of a Function

A function can be visualized as a machine that takes an input value, processes it according to a rule, and produces an output value.

• Input: The independent variable (x)

• Output: The dependent variable (f(x))

Examples of Functions

• Example 1: Express the function (or rule) in words.

  • a)

  • b)

  Explanation: For , multiply the input by 3 and add 2. For , add 2 to the input, then multiply by 3.

• Example 3: Let

  • a) Evaluate

  • b) Evaluate

  • c) Evaluate

  Explanation: Substitute the given value for into the function and simplify.

Net Change of a Function

Net change refers to the change in the value of a function as the input changes from one value to another. It is useful for understanding how a function behaves over an interval.

• The net change in the value of a function from to is given by , where .

• If you have two points and want to know the net change, subtract the function values: .

• Example: Find the net change in the value of the function between and .

Difference Quotient

The difference quotient is a formula used to compute the average rate of change of a function over an interval. It is foundational for calculus.

• For a function , the difference quotient is: , where

• Example 5: For :

  • a) Find

  • b) Find

  • c) Find the difference quotient

Domain of a Function

Definition and Examples

The domain of a function is the set of all values of for which is real and defined. Restrictions may arise from division by zero or taking the square root of a negative number.

• Example: Find the domain for each function below:

  • a)

  • b)

  • c)

  • d)

  • e)

  Explanation:

  • For rational functions, exclude values that make the denominator zero.

  • For square root functions, require the radicand to be non-negative.

  • For functions with both, apply both restrictions.

Table: Domain Restrictions

Summary

• Functions assign each input exactly one output.

• Evaluate functions by substituting values for the independent variable.

• Net change measures the difference in function values over an interval.

• The difference quotient is key for understanding rates of change.

• The domain is determined by the set of allowable input values.