Section 1: Relations and Functions

Part 1: Relations

A relation is a correspondence between two variables: a set X, called the domain, and a set Y, called the range. In a relation, each element from the domain corresponds to at least one element from the range. The domain is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• Verbal Description: Describes the relationship in words.

• Numerical Description: Uses a table or list of ordered pairs to show the relation.

• Graphical Description: Plots the ordered pairs on a coordinate plane or uses a mapping diagram.

• Algebraic Description: Expresses the relation using an equation.

Example: An astronaut throws a rock on the Moon and measures its height at various times. The relation between time and height can be described in several ways:

The same relation can be represented graphically or as a mapping:

The algebraic model for this relation is $y = 20 - 0.8x^2$.

Domain: {0, 1, 2, 2.5, 3, 4, 5} Range: {20, 19.2, 16.8, 15, 12.8, 7.2, 0}

Part 2: Functions

A function is a special type of relation in which each element of the domain corresponds to exactly one element of the range. All functions are relations, but not all relations are functions.

• To determine if a relation is a function, check that each input (x-value) is paired with only one output (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.

Examples:

• (a) $y = x^2$ is a function (passes the vertical line test).

• (b) $y = x^3$ is a function.

• (c) $x = y^2$ is not a function (fails the vertical line test).

• (d) $x^2 + y^2 = 1$ is not a function (circle, fails the vertical line test).

Part 3: Function Notation

Functions are often labeled with letters such as f and g. If $f$ is a function, then for each $x$ in the domain, the corresponding $y$ in the range is written as $f(x)$, read as "f of x." This is called function notation.

• $y = f(x)$ means "y is a function of x" or "y depends on x."

• To evaluate a function at a particular value, substitute that value for $x$ in the function's equation.

Example: If $f(x) = x^2 - 2x + 1$, then $f(2) = 2^2 - 2 \times 2 + 1 = 1$.

Part 4: Domain of Functions

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

• Polynomial functions: Domain is all real numbers, $(-\infty, \infty)$.

• Rational functions: Domain excludes values that make the denominator zero.

• Radical functions (even roots): Domain is where the radicand is greater than or equal to zero.

Example: For $f(x) = \sqrt{x+2}$, set $x+2 \geq 0 \implies x \geq -2$. So, domain is $[-2, \infty)$.

Part 5: Function Equations and Intercepts

From the equation of a function, you can find:

• Specific values: Substitute a value for $x$ to find $f(x)$.

• x-intercepts (zeros): Set $f(x) = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $f(0)$.

Example: For $f(x) = \frac{x+3}{x+4}$:

• Domain: $x \neq -4$

• x-intercept: Set $x+3=0 \implies x=-3$

• y-intercept: $f(0) = \frac{3}{4}$

Section 2: The Graph of a Function

Understanding Graphs

The graph of a function $y = f(x)$ is the set of all ordered pairs $(x, y)$ plotted on a coordinate plane. From the graph, you can determine:

• Domain and Range: The set of all $x$-values and $y$-values represented on the graph.

• Intercepts: Where the graph crosses the axes.

• Intervals of Positivity/Negativity: Where $f(x) > 0$ (above x-axis) or $f(x) < 0$ (below x-axis).

• Increasing/Decreasing/Constant Intervals: Where the function rises, falls, or stays the same as $x$ increases.

• Local and Absolute Extrema: Highest or lowest points in a region (local) or on the entire domain (absolute).

Section 3: Graphing Functions

Part 1: Library of Functions

A family of functions is a group of functions with similar shapes and properties. The parent function is the simplest form in the family. Common parent functions include:

Part 2: Graphing Functions Using Transformations

Functions can be graphed by applying transformations to their parent functions. Transformations include:

• Vertical Shifts: $y = f(x) + k$ shifts the graph up by $k$ units; $y = f(x) - k$ shifts it down by $k$ units.

• Horizontal Shifts: $y = f(x - h)$ shifts the graph right by $h$ units; $y = f(x + h)$ shifts it left by $h$ units.

• Vertical Stretch/Compression: $y = a f(x)$ stretches the graph vertically if $a > 1$ and compresses it if $0 < a < 1$.

• Horizontal Stretch/Compression: $y = f(ax)$ compresses the graph horizontally if $a > 1$ and stretches it if $0 < a < 1$.

• Reflections: $y = -f(x)$ reflects the graph over the x-axis; $y = f(-x)$ reflects it over the y-axis.

Summary Table of Transformations:

Example: To graph $g(x) = x^2 + 3$, start with the parent function $f(x) = x^2$, then shift the graph up by 3 units.

Additional info: These foundational concepts are essential for understanding more advanced topics in College Algebra, such as polynomial, rational, exponential, and logarithmic functions.