Functions, Domains, Ranges, and Inverses: Calculus Study Notes

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 $f(x)$, where $x$ is the input variable.

Example: Quadratic Function

Consider the function $f(x) = -x^2 + 4$.

• Graph: The graph is a downward-opening parabola with vertex at $(0, 4)$.

• Domain: $(-\infty, \infty)$ (all real numbers)

• Range: $(-\infty, 4]$ (all real numbers less than or equal to 4)

• Is it a function? Yes, because each $x$ value has only one $y$ 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 $x$-values covered by the graph.

• Range: The set of $y$-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 $f(x)$ as $x$ approaches a value $a$ is the value that $f(x)$ approaches as $x$ gets arbitrarily close to $a$. It is written as $\lim_{x \to a} f(x)$.

• Left-hand limit: $\lim_{x \to a^-} f(x)$ (as $x$ approaches $a$ from the left)

• Right-hand limit: $\lim_{x \to a^+} f(x)$ (as $x$ approaches $a$ from the right)

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

Continuity

A function $f(x)$ is continuous at $x = a$ if:

• $f(a)$ is defined

• $\lim_{x \to a} f(x)$ exists

• $\lim_{x \to a} f(x) = f(a)$

If any of these conditions fail, the function is discontinuous at $x = a$.

Example: Evaluating Limits and Continuity from a Graph

• To find $\lim_{x \to a^-} f(x)$, observe the $y$-value as $x$ approaches $a$ from the left.

• To find $\lim_{x \to a^+} f(x)$, observe the $y$-value as $x$ approaches $a$ from the right.

• To check continuity at $x = a$, ensure the function is defined at $a$ and both one-sided limits are equal to $f(a)$.

Inverse Functions

Definition and Properties

An inverse function $f^{-1}(x)$ reverses the effect of the original function $f(x)$. If $f(a) = b$, then $f^{-1}(b) = a$.

• 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 $f^{-1}(x)$ is the reflection of the graph of $f(x)$ across the line $y = x$.

Example: Inverse of $y = x^3$

• The function $y = x^3$ is one-to-one and has an inverse $y = \sqrt[3]{x}$.

• The graphs of $y = x^3$ and $y = \sqrt[3]{x}$ are symmetric with respect to the line $y = x$.

Derivatives of Inverse Functions

If $y = f(x)$ is a one-to-one differentiable function with inverse $x = f^{-1}(y)$, then:

• The derivative of the inverse function is given by:

$\frac{d}{dy} f^{-1}(y) = \frac{1}{f'(x)}$ where $x = f^{-1}(y)$

• 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

Property	Definition	How to Determine
Domain	Set of all possible input values	Look for $x$-values where the function is defined
Range	Set of all possible output values	Find all $y$-values the function attains
Function?	Each input has one output	Vertical line test
Inverse?	Inverse is also a function	Horizontal line test
Continuity	No breaks, jumps, or holes	Check limits and function value at a point

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.