Unit 1: Relations and Functions

This unit bridges the gap between basic numerical patterns and advanced mathematical modeling by formalizing the concept of relations and functions. Students will master the use of the Cartesian coordinate system to visualize mathematical dependencies and solve complex problems involving linear and quadratic models. By exploring properties such as injectivity, surjectivity, and bijectivity, this chapter provides the foundational logic required for all subsequent studies in calculus and higher-order algebra .

1.1 Relations

• In a relation, two things are related to each other by a relating phrase.
• Mathematically, a relation is defined as a set of ordered pairs.
• The Cartesian coordinate system in two dimensions (rectangular coordinate system) is defined by an ordered pair of perpendicular lines called axes.
• The point where these axes meet is the origin, which serves as the zero point for both number lines.
• The domain of a relation is the set consisting of all the first components of the ordered pairs.
• The range of a relation is the set consisting of all the second components of the ordered pairs.
• Graphs of relations are created by locating ordered pairs as points in a coordinate system.

Process: Sketching Graphs for Relations Involving Inequalities

1. Draw the graph of the line(s) or curve(s) defining the boundary in the relation on the coordinate system.
2. If the relating inequality is $\le$ or $\ge$, use a solid line to show the boundary is part of the solution.
3. If the inequality is < or >, use a broken line (dashed line).
4. Select an arbitrary ordered pair (test point) from one of the regions created.
5. Substitute the test point into the inequality; if the statement is true, shade that region; if false, shade the opposite region .

1.2 Functions

• A function is a special type of relation in which no two distinct ordered pairs have the same first element.
• Terms such as map or mapping are also used to denote a function.
• The domain of a function is the entire set of possible values for the independent variables.
• The co-domain is the set containing all potential outputs, while the range is the set of actual outputs produced after substituting the domain.
• For any x in the domain, the corresponding element f(x) in the range is the image of x, and x is the pre-image of f(x).
• The functional value of f at x is the output f(x) obtained after evaluating the function at x.
• The vertical line test determines whether a graph is a function; it is a function if and only if no vertical line intersects the graph at more than one point.

1.2.2 Combination of Functions

• Two functions can be combined by arithmetic operations to form a new combination of functions.
• The sum of functions is defined as $(f+g)(x) = f(x) + g(x)$.
• The difference of functions is defined as $(f-g)(x) = f(x) – g(x)$.
• The product of functions is defined as $(f \cdot g)(x) = f(x) \cdot g(x)$.
• The quotient of functions is defined as $(f \div g)(x) = f(x) \div g(x)$, with the restriction that $g(x) \neq 0$.
• The domain of these combinations consists of real numbers common to the domains of both f and g.

1.2.3 Types of Functions

• A one-to-one function (injective function) exists if each element in the domain has a distinct image in the co-domain.
• An onto function (surjective function) exists if each element in the co-domain has at least one pre-image in the domain.
• A one-to-one correspondence (bijective) occurs when a function is both one-to-one and onto.

1.2.4 Graphs of Functions

• A linear function is defined by f(x) = ax + b (where $a \neq 0$), and its graph is a straight line.
• The constant a represents the slope and b is the y-intercept.
• A constant function is defined by f(x) = b; its domain is all real numbers and its range is the single value $\{b\}$.
• A quadratic function is defined by $f(x) = ax^2 + bx + c$ (where $a \neq 0$); its graph is a U-shaped curve called a parabola.
• The number a is the leading coefficient of the function.
• All parabolas are symmetric with respect to a line called the axis of symmetry.
• The vertex is the point on the coordinate system where the graph turns either upward or downward
• This vertex is also known as the turning point of the graph.
• If $a > 0$, the parabola opens upward and the vertex provides a minimum value.
• If $a < 0$, the parabola opens downward and the vertex provides a maximum value.
• The x-intercept is the point where the graph crosses the x-axis (y = 0).
• The y-intercept is the point where the graph crosses the y-axis (x = 0).

Process: Solving Quadratic Equations Graphically

1. Rearrange the equation so that one side is equal to zero, forming the function f(x) = 0.
2. Draw the graph of the quadratic function.
3. Read the x-coordinate(s) of the point(s) where the curve crosses the x-axis; these are the solutions.

Process: Solving Quadratic Inequalities Graphically

1. Write the quadratic inequality in standard form ($ax^2 + bx + c \ge 0$ or $\le 0$).
2. Graph the quadratic function $f(x) = ax^2 + bx + c$ using properties like the vertex and intercepts.
3. If seeking values where f(x) < 0, identify the region where the parabola is below the x-axis.
4. If seeking values where f(x) > 0, identify the region where the parabola is above the x-axis.

1.3 Applications of Relations and Functions

• Relations model real-world dependencies, such as the relationship between a woman’s femur length and her height used in forensic science.
• Linear functions are used to calculate service costs, such as electrician fees based on hours worked ($C(t) = 150t + 70$).
• Quadratic functions model the motion of objects thrown into the air, reaching a peak at the maximum value before falling.

Key Terminology

• Axis (orthogonal-axis): One of the perpendicular lines defining a coordinate system.
• Axis of symmetry: The vertical line that passes through the vertex of a parabola.
• Combination of functions: A new function formed by performing arithmetic operations on two or more functions.
• Constant function: A function that can be written in the form f(x) = b.
• Coordinate system: A system defined by an ordered pair of perpendicular lines used to represent ordered pairs graphically.
• Domain: The set of all first components in the ordered pairs of a relation or the valid input set for a function.
• Function: A special type of relation where each input is related to exactly one output.
• Leading coefficient: The coefficient a of the highest-degree term in a polynomial function.
• Linear function: A function written as f(x) = ax + b, where $a \neq 0$.
• Parabola: The U-shaped curve representing the graph of a quadratic function.
• Quadratic function: A function written as $f(x) = ax^2 + bx + c$, where $a \neq 0$.
• Range: The set of all second components in the ordered pairs of a relation or the set of output values for a function.
• Relation: A set of ordered pairs linking objects or values.
• Slope: A constant a that determines the steepness and direction of a linear function.
• Turning point: The point on a coordinate system where a quadratic function turns either upward or downward.
• Vertex: The intersection point of a parabola and its axis of symmetry.
• X-intercept: The coordinate where a graph crosses the horizontal axis.
• Y-intercept: The coordinate where a graph crosses the vertical axis.