Functions: Definitions, Properties, and Applications

Functions and Relations

Definition and Concept of a Function

A function is a mathematical rule that assigns to each element in a set, called the domain, exactly one element in another set, called the range. Functions are used to describe how one quantity depends on another, such as production cost depending on items produced or velocity depending on time.

• Mapping: Each element in the domain is mapped to one and only one element in the range.

• Rule: The function's rule is the method or formula used for assigning values.

• Domain: The set of input values.

• Range: The set of output values.

Examples of functions in real life:

• Production cost as a function of items produced

• Revenue as a function of items sold

• Blue book value of a car as a function of odometer reading

• Velocity of a car as a function of time

Functions vs. Relations

A relation is a set of ordered pairs. Every function is a relation, but not every relation is a function. A function is a relation in which no two ordered pairs have the same first coordinate and different second coordinates.

• Function: {(−1, 2), (0, 5), (2, 6), (4, 5)} is a function because each input has only one output.

• Not a function: {(−3, 2), (1, −1), (0, 3), (1, 2)} is not a function because the input 1 maps to both −1 and 2.

Graphical Identification of Functions

Vertical Line Test

The Vertical Line Test is a graphical method to determine if a relation is a function. A graph represents a function if and only if no vertical line crosses the graph more than once.

• If a vertical line intersects the graph at more than one point, the relation is not a function.

• If every vertical line intersects the graph at most once, the relation is a function.

Example: The graph below can be analyzed using the vertical line test to determine if it represents a function.

Identifying Functions from Equations

Equations and Functions

Not all equations in two variables define functions. To determine if an equation defines y as a function of x, check if each x-value corresponds to only one y-value.

• Example: $|y| = x$ is not a function because for some x-values, there are two possible y-values.

• Example: $y = x^2 - 3x + 2$ is a function because each x-value yields only one y-value.

• Example: $x^2 + y^2 = 1$ is not a function because some x-values correspond to two y-values.

• Example: $y^2 = 2x + 1$ is not a function for the same reason.

Domain and Range

Definitions

The domain of a relation is the set of all first coordinates (inputs), and the range is the set of all second coordinates (outputs).

• Example: For the relation {(−1, 2), (0, 5), (2, 6), (4, 5)}, the domain is {−1, 0, 2, 4} and the range is {2, 5, 6}.

• Example: For the relation {(-1,1), (3,9), (3,-9)}, the domain is {−1, 3} and the range is {1, 9, −9}. This is not a function because 3 maps to both 9 and −9.

When using equations to define functions, the domain is often assumed to be all real numbers for which the expression is defined (implied domain).

• Undefined expressions: Division by zero and even roots of negative numbers.

• Example: Find the domain of $y = \frac{1}{x^2 - 4}$: The domain excludes $x = 2$ and $x = -2$.

• Example: Find the domain of $y = \sqrt{2x - 4}$: The domain is $x \geq 2$.

Function Notation and Evaluation

Function Notation

Functions are often written using special notation. Instead of $y = x^2 + 2x + 1$, we write $f(x) = x^2 + 2x + 1$. The variable is a placeholder for values in the domain.

• Example: $f(t) = t^2 + 2t + 1$

• The variable is called a "dummy" variable.

Evaluating Functions

To evaluate a function for a specific value, substitute the value into the function's rule.

• Example: $f(x) = x^2 + 2x + 1$

• $f(0) = 0^2 + 2 \cdot 0 + 1 = 1$

• $f(1) = 1^2 + 2 \cdot 1 + 1 = 4$

• $f(-1) = (-1)^2 + 2 \cdot (-1) + 1 = 0$

• $f(a) = a^2 + 2a + 1$

• $f(\sqrt{a}) = (\sqrt{a})^2 + 2 \sqrt{a} + 1 = a + 2 \sqrt{a} + 1$

• $f(x + h) = (x + h)^2 + 2(x + h) + 1$

Example with a set of ordered pairs: Let $h = \{(1,4), (6,0), (7,9)\}$

• $h(7) = 9$

• $h(0)$ is undefined (0 is not in the domain).

• When $h(x) = 0$, $x = 6$.