Functions, Graphs, and Circles: Precalculus Study Notes

Functions and Their Graphs

Relations and Functions

In mathematics, a relation is a connection between input and output values, typically represented as ordered pairs (x, y). A function is a special type of relation in which each input (x-value) has at most one output (y-value).

• Relation: Any set of ordered pairs (x, y).

• Function: A relation where each x-value is paired with only one y-value.

• Vertical Line Test: A graph represents a function 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 x-value is unique. The set {(2, 5), (0, 2), (2, 9)} is not a function because x = 2 is paired with two different y-values.

Verifying Functions from Equations

To determine if an equation defines y as a function of x, solve for y and check if each x-value yields only one y-value.

• If y appears with an even power (e.g., y2), the equation may not define a function.

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

• Example: y + 4 = 3x is a function; x2 + y2 = 25 is not a function.

Domain and Range

Finding Domain and Range from Graphs

The domain of a function is the set of all possible x-values, and the range is the set of all possible y-values. To find these from a graph, project the graph onto the x-axis (domain) and y-axis (range).

• Domain: Allowed x-values.

• Range: Allowed y-values.

• Use interval notation for continuous intervals and the union symbol (∪) for multiple intervals.

• Brackets [ ] mean the value is included; parentheses ( ) mean the value is not included.

Finding Domain from Equations

When given an equation, determine the domain by identifying restrictions:

• For square roots: x-values must make the radicand non-negative.

• For denominators: x-values must not make the denominator zero.

• Example: For f(x) = \frac{2}{x-5}, the domain is all real numbers except x = 5.

• Example: For f(x) = \sqrt{x}, the domain is x ≥ 0.

Common Functions and Their Graphs

Types of Functions

Several basic functions frequently appear in precalculus:

• Constant Function: f(x) = c. Domain: (−∞, ∞). Range: {c}.

• Identity Function: f(x) = x. Domain: (−∞, ∞). Range: (−∞, ∞).

• Square Function: f(x) = x2. Domain: (−∞, ∞). Range: [0, ∞).

• Cube Function: f(x) = x3. Domain: (−∞, ∞). Range: (−∞, ∞).

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

• Cube Root Function: f(x) = \sqrt[3]{x}. Domain: (−∞, ∞). Range: (−∞, ∞).

Transformations of Functions

Types of Transformations

Transformations change 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: Moves the graph horizontally or vertically.

• Stretch/Shrink: Changes the graph's size by multiplying by a constant.

• Reflection: $-f(x)$ reflects over the x-axis; $f(-x)$ reflects over the y-axis.

• Shift: $f(x-h) + k$ shifts right by h and up by k.

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

Domain and Range of Transformed Functions

Transformations can affect the domain and range. Analyze the new graph to determine these sets.

Function Operations

Adding, Subtracting, Multiplying, and Dividing Functions

Functions can be combined using addition, subtraction, multiplication, and division. The domain of the resulting function is the intersection of the domains of the original functions, with additional restrictions for division (denominator ≠ 0).

• Addition: $(f+g)(x) = f(x) + g(x)$

• Subtraction: $(f-g)(x) = f(x) - g(x)$

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

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

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 excludes x-values not defined for g(x) and those that make f(g(x)) undefined.

• Example: If $f(x) = x^2$ and $g(x) = x-1$, then $(f \circ g)(x) = (x-1)^2$.

Function Decomposition

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

• Example: $h(x) = \sqrt{x-2}$ can be written as $f(g(x))$ where $g(x) = x-2$ and $f(x) = \sqrt{x}$.

Circles in the Coordinate Plane

Standard Form of a Circle

The equation of a circle in standard form is $(x-h)^2 + (y-k)^2 = r^2$, where (h, k) is the center and r is the radius. All points on the circle are a fixed distance r from the center.

• Example: $(x-1)^2 + (y-2)^2 = 9$ has center (1, 2) and radius 3.

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

General Form to Standard Form

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

• Group x and y terms, move the constant to the right.

• Add the necessary values to complete the square for both x and y.

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

Practice Problems

• Given $x^2 + y^2 - 2x + 4y - 4 = 0$, complete the square to find the center and radius.

• Given $x^3 + y^2 + 4x - 8y + 4 = 0$, determine if the equation represents a circle.

Additional info: Images included are directly relevant to the explanation of domain, range, transformations, and circles as required by the study notes.