Functions and Their Properties

Definition of a Function

A function is a relation that assigns to each element in the domain exactly one element in the range. Functions are fundamental objects in calculus, used to model relationships between varying quantities.

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Range: The set of all possible output values (y-values) that the function can produce.

• Notation: Functions are often written as , where is the input variable.

Example: Quadratic Function

Consider the function .

• Graph: The graph is a downward-opening parabola with vertex at .

• Domain: (all real numbers)

• Range: (all real numbers less than or equal to 4)

• Is it a function? Yes, because each value has only one value.

• Inverse: The function does not have an inverse that is also a function over its entire domain, because it fails the horizontal line test.

Example: Function Defined by a Graph

Given a graph, you can determine:

• Domain: The set of -values covered by the graph.

• Range: The set of -values the graph attains.

• Is it a function? Use the vertical line test: if any vertical line crosses the graph more than once, it is not a function.

• Inverse: Use the horizontal line test: if any horizontal line crosses the graph more than once, the inverse is not a function.

Limits and Continuity

Definition of a Limit

The limit of a function as approaches a value is the value that $f(x)$ approaches as $x$ gets arbitrarily close to $a$. It is written as .

• Left-hand limit: (as approaches from the left)

• Right-hand limit: (as approaches from the right)

• Limit exists: Only if both left and right limits exist and are equal.

Continuity

A function is continuous at if:

• is defined

• exists

If any of these conditions fail, the function is discontinuous at .

Example: Evaluating Limits and Continuity from a Graph

• To find , observe the -value as approaches from the left.

• To find , observe the -value as approaches from the right.

• To check continuity at , ensure the function is defined at and both one-sided limits are equal to .

Inverse Functions

Definition and Properties

An inverse function reverses the effect of the original function . If , then .

• Existence: A function has an inverse that is also a function if and only if it is one-to-one (passes the horizontal line test).

• Graphical Relationship: The graph of is the reflection of the graph of across the line .

Example: Inverse of

• The function is one-to-one and has an inverse .

• The graphs of and are symmetric with respect to the line .

Derivatives of Inverse Functions

If is a one-to-one differentiable function with inverse , then:

• The derivative of the inverse function is given by:

where

• This means the slope of the inverse at a point is the reciprocal of the slope of the original function at the corresponding point.

Table: Function Properties Summary

Additional info:

• Some handwritten notes and graphs were interpreted based on standard calculus curriculum and context.

• Examples and explanations were expanded for clarity and completeness.