Functions and Graphs

Relations and Functions

In algebra, 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 where each input (x) has at most one output (y). This means that for every x-value, there is only one corresponding y-value.

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point. If a vertical line crosses the graph more than once, the relation is not a function.

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

Verifying Functions from Equations

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

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

• Example: y + 4 = 3x is a function; x2 + y2 = 25 is not a function because for some x-values, there are two possible y-values.

Domain and Range of a Graph

The domain of a function is the set of all possible input (x) values, and 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. Use interval notation or set builder notation to express these sets.

• Interval Notation: Uses brackets [ ] for inclusive and parentheses ( ) for exclusive endpoints.

• Set Builder Notation: Describes the set using inequalities.

Common Functions and Their Graphs

Types of Common Functions

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

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

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

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

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

• Cube Root Function: f(x) = ∛x. Domain and Range: (−∞, ∞).

Equations of Two Variables

Graphing and Satisfying Equations

Equations with two variables (x and y) can be graphed by plotting all points (x, y) that satisfy the equation. If a point satisfies the equation, it lies on the graph.

• Example: The equation x + y = 5 is a line. Points like (3, 2) and (4, 1) satisfy the equation and are on the line.

Graphing by Plotting Points

1. Isolate y on one side: y = ...

2. Choose 3–5 x-values and calculate corresponding y-values.

3. Plot the (x, y) points.

4. Connect the points with a line or curve.

Lines: Slope and Equations

Slope of a Line

The slope (m) measures the steepness of a line and is calculated as the change in y divided by the change in x between two points:

• Positive Slope: Line rises from left to right.

• Negative Slope: Line falls from left to right.

• Zero Slope: Horizontal line.

• Undefined Slope: Vertical line.

Forms of Linear Equations

• Slope-Intercept Form:

• Point-Slope Form:

• Standard Form:

Parallel and Perpendicular Lines

• Parallel Lines: Have equal slopes ().

• Perpendicular Lines: Slopes are negative reciprocals ().

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 (h) or vertically (k).

• Stretch/Shrink: Multiplies the function by a constant to stretch or compress it vertically or horizontally.

Function Operations

Adding, Subtracting, Multiplying, and Dividing Functions

• Addition/Subtraction: Combine like terms. The domain is the intersection of the domains of the original functions.

• Multiplication: Multiply the functions. The domain is the intersection of the domains of the original functions.

• Division: Divide the functions, excluding values where the denominator is zero.

Function Composition and Decomposition

Function Composition

Function composition involves substituting one function into another: . The domain of the composite function is the set of all x-values for which g(x) is in the domain of f.

Function Decomposition

Decomposition is the reverse process: expressing a function as the 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:

• Center: (h, k)

• Radius: r

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

General Form to Standard Form

To convert a general form equation to standard form, complete the square for both x and y terms.

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

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

• Rewrite as .

Summary Table: Types of Transformations