Functions and Their Graphs: Foundations, Transformations, and Circles

Functions and Their Graphs

Relations and Functions

Understanding the distinction between relations and functions is fundamental in precalculus. A relation is any set of ordered pairs (x, y), representing a connection between input (x) and output (y) values. A function is a special type of relation in which each input is associated with at most one output.

• Relation: Any pairing of x-values (inputs) with y-values (outputs).

• Function: Each x-value is paired with only one y-value.

• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Example: The set {(−3, 5), (0, 2), (3, 5)} is a function because each input has only one output. The set {(2, 5), (0, 2), (2, 9)} is not a function because the input 2 is paired with two different outputs (5 and 9).

Verifying Functions from Equations and Graphs

To determine if an equation defines a function, solve for y in terms of x. If each x produces only one y, the equation is a function. If y is raised to an even power (e.g., y2), the equation typically does not define y as a function of x.

• Function Notation: If y = 3x − 4, we can write f(x) = 3x − 4.

• Graphical Test: Use the vertical line test to check if a graph represents a function.

Domain and Range

Finding the Domain and Range of a Graph

The domain of a function is the set of all possible input (x) values, while the range is the set of all possible output (y) values. To find the domain, project the graph onto the x-axis; for the range, project onto the y-axis.

• Interval Notation: Uses parentheses and brackets to describe intervals (e.g., [−4, 5)).

• Set Builder Notation: Describes the set using inequalities (e.g., {x | −4 ≤ x < 5}).

• Union Symbol (∪): Used when the domain or range consists of multiple intervals.

• Inclusion: [ ] or ≤, ≥ means the endpoint is included; ( ) or <, > means it is not included.

Finding the Domain of an Equation

When given an equation, determine the domain by identifying values that make the function undefined. Common restrictions include:

• Square Roots: The expression inside must be non-negative (≥ 0).

• Denominators: The denominator cannot be zero.

Example: For $f(x) = \sqrt{x}$, the domain is $[0, \infty)$. For $f(x) = \frac{2}{x-5}$, the domain is $(-\infty, 5) \cup (5, \infty)$.

Common Functions and Their Graphs

Basic Function Types

• Constant Function: $f(x) = c$ (Domain: $(-\infty, \infty)$, Range: $\{c\}$)

• Identity Function: $f(x) = x$ (Domain and Range: $(-\infty, \infty)$)

• Square Function: $f(x) = x^2$ (Domain: $(-\infty, \infty)$, Range: $[0, \infty)$)

• Cube Function: $f(x) = x^3$ (Domain and Range: $(-\infty, \infty)$)

• Square Root Function: $f(x) = \sqrt{x}$ (Domain: $[0, \infty)$, Range: $[0, \infty)$)

• Cube Root Function: $f(x) = \sqrt[3]{x}$ (Domain and Range: $(-\infty, \infty)$)

Transformations of Functions

Types of Transformations

Transformations alter the position or shape of a function's graph. The main types are:

• Reflection: Flips the graph over the x-axis or y-axis.

• Shift (Translation): Moves the graph horizontally and/or vertically.

• Stretch/Shrink: Changes the graph's steepness or width by multiplying by a constant.

Reflections

• Over the x-axis: $y = -f(x)$ (y-values change sign)

• Over the y-axis: $y = f(-x)$ (x-values change sign)

Shifts

• Vertical Shift: $f(x) + k$ shifts up by $k$; $f(x) - k$ shifts down by $k$.

• Horizontal Shift: $f(x - h)$ shifts right by $h$; $f(x + h)$ shifts left by $h$.

Stretches and Shrinks

• Vertical Stretch/Shrink: $c \cdot f(x)$ stretches if $c > 1$, shrinks if $0 < c < 1$.

• Horizontal Stretch/Shrink: $f(c \cdot x)$ shrinks if $c > 1$, stretches if $0 < c < 1$.

Domain and Range of Transformed Functions

Transformations can affect the domain and range. Analyze the new graph or apply the transformation rules to the original domain and range.

Function Operations

Adding, Subtracting, Multiplying, and Dividing Functions

• Add/Subtract: $(f \pm g)(x) = f(x) \pm g(x)$

• Multiply: $(f \cdot g)(x) = f(x) \cdot g(x)$

• Divide: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

• Domain: The intersection of the domains of $f$ and $g$, with additional restrictions for division (denominator cannot be zero).

Function Composition and Decomposition

Function Composition

Function composition involves substituting one function into another: $(f \circ g)(x) = f(g(x))$. The domain of the composite function consists of all x-values for which $g(x)$ is defined and $f(g(x))$ is also defined.

• Evaluating: Compute $g(x)$, then substitute into $f$.

• Domain: Exclude x-values not in the domain of $g$ or for which $g(x)$ is not in the domain of $f$.

Function Decomposition

Decomposition is the reverse process: expressing a function as a composition of two or more simpler functions.

Circles in the Coordinate Plane

Standard Form of a Circle

The equation of a circle with center $(h, k)$ and radius $r$ is:

$(x - h)^2 + (y - k)^2 = r^2$

• Center: $(h, k)$

• Radius: $r$

• A circle is not a function because it fails the vertical line test.

Graphing Circles

• Identify the center and radius from the equation.

• Plot the center, then mark points a distance $r$ in all four directions (up, down, left, right).

• Connect these points with a smooth curve to complete the circle.

General Form to Standard Form

To convert a general form equation $x^2 + y^2 + Ax + By + C = 0$ to standard form, complete the square for both $x$ and $y$ terms.

• Group $x$ and $y$ terms, move the constant to the other side.

• Add the necessary values to complete the square for each variable.

• Rewrite as $(x - h)^2 + (y - k)^2 = r^2$.

Practice and Applications

• Given an equation, determine if it represents a circle and, if so, find its center and radius.

• Write the equation of a circle given its center and radius.

Additional info: These notes cover the foundational aspects of functions, their graphs, transformations, operations, composition, and the geometry of circles, as outlined in a typical precalculus curriculum.