Linear Function

Solving Linear Equations

• Linear equations involve variables to the first power.
• The goal is to isolate the variable on one side of the equation.
• Use inverse operations to isolate the variable.
• Addition and subtraction are inverse operations.
• Multiplication and division are inverse operations.
• Apply operations to both sides of the equation to maintain equality.

Steps for Solving Linear Equations

• Simplify both sides of the equation by combining like terms and distributing if necessary.
• Use addition or subtraction to move all terms with the variable to one side of the equation.
• Use addition or subtraction to move all constant terms to the other side of the equation.
• Divide or multiply to solve for the variable.
• Check the solution by substituting it back into the original equation.

Example of Solving a Linear Equation

• Solve: 3x + 5 = 14
• Subtract 5 from both sides: 3x + 5 - 5 = 14 - 5, which simplifies to 3x = 9.
• Divide both sides by 3: (3x)/3 = 9/3, which simplifies to x = 3.
• Check: 3(3) + 5 = 9 + 5 = 14, so the solution is correct.

Word Problems Related to Linear Equations

• Word problems require translating real-world scenarios into mathematical equations.
• Identify the unknown variable.
• Define the variable
• Assign a variable to represent the unknown quantity (e.g., x, y).
• Translate the information given in the problem into an equation.
• Look for key words that indicate mathematical operations
• "Sum" or "total" suggests addition.
• "Difference" suggests subtraction.
• "Product" suggests multiplication.
• "Quotient" suggests division.
• "Is," "equals," or "results in" suggests equality.

Common Types of Word Problems

• Age problems: Involve finding the ages of people based on given relationships between their ages.
• Mixture problems: Involve combining two or more mixtures with different concentrations to create a new mixture.
• Rate problems: Involve distance, rate, and time, often using the formula distance = rate × time.
• Number problems: Involve finding unknown numbers based on given conditions.

Example of a Word Problem

• Problem: John is twice as old as his sister. In 5 years, he will be 8 years older than her. How old is his sister now?
• Let 'x' be the sister's current age.
• John's current age is 2x.
• In 5 years, the sister's age will be x + 5.
• In 5 years, John's age will be 2x + 5.
• Equation: 2x + 5 = (x + 5) + 8
• Simplify: 2x + 5 = x + 13
• Subtract x from both sides: x + 5 = 13
• Subtract 5 from both sides: x = 8
• The sister's current age is 8 years old.
• Check: John's current age is 2 * 8 = 16. In 5 years, the sister will be 13 and John will be 21, which is 8 years older than the sister.

Strategies for Solving Word Problems

• Read the problem carefully: Understand what the problem is asking.
• Identify the knowns and unknowns: Determine what information is given and what you need to find.
• Assign variables: Choose variables to represent the unknown quantities.
• Write an equation: Translate the information into a mathematical equation.
• Solve the equation: Use algebraic techniques to find the value of the variable.
• Check the solution: Make sure the solution makes sense in the context of the problem.
• Label the answer: Include appropriate units (e.g., years, meters, kilograms).

Functions

• A function is a relation in which each input (x-value) has exactly one output (y-value).
• Functions can be represented in multiple ways: equations, graphs, tables, or words.
• The vertical line test is used to determine if a graph represents a function.
• A graph represents a function if no vertical line intersects the graph more than once

Representations of Functions

• Equation: y = f(x), where x is the input and f(x) is the output.
• Graph: A visual representation of the function on a coordinate plane.
• Table: A table of values showing the inputs and their corresponding outputs.
• Words: A description of the relationship between the inputs and outputs.

Function Notation

• f(x) is the notation used to represent a function.
• "f(x)" is read as "f of x"
• f(x) represents the output of the function f for a given input x.
• To evaluate a function, substitute the input value for x in the function's equation.
• For example, if f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11.

Domain and Range

• Domain: The set of all possible input values (x-values) for which the function is defined.
• Range: The set of all possible output values (y-values) that the function can produce.
• The domain and range can be expressed in interval notation, set notation, or graphically.

Types of Functions

• Linear functions: Functions with a constant rate of change, represented by a straight line on a graph (y = mx + b).
• Quadratic functions: Functions with a squared term, represented by a parabola on a graph (y = ax^2 + bx + c).
• Exponential functions: Functions where the variable appears in the exponent (y = a^x).
• Polynomial functions: Functions involving non-negative integer powers of x.

Relations

• A relation is any set of ordered pairs (x, y).
• A relation does not necessarily have the property that each input has a unique output.
• Every function is a relation, but not every relation is a function.

Examples of Relations that are Not Functions

• A set of ordered pairs where one x-value is associated with multiple y-values; for example, {(1, 2), (1, 3), (2, 4)} is a relation but not a function because the input 1 has two different outputs (2 and 3).
• An equation like x = y^2 is a relation but not a function because for a single x-value, there can be two y-values (e.g., if x = 4, y can be 2 or -2).