Equations Basics

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Questions and Answers

What is the main goal when solving an equation?

  • To simplify the equation
  • To find the value of the variable that makes the equation true (correct)
  • To isolate the variable
  • To find the value of the constant

What type of equation is 2x^2 + 3x - 4 = 0?

  • Linear equation
  • Polynomial equation
  • Quadratic equation (correct)
  • Exponential equation

What is the domain of the function f(x) = 1/x?

  • All real numbers
  • All real numbers except 0 (correct)
  • Only positive real numbers
  • All real numbers except 1

What is the graph of a function?

<p>A visual representation of the relationship between the input and output values (C)</p> Signup and view all the answers

What is the purpose of the distributive property in solving equations?

<p>To expand and simplify expressions (B)</p> Signup and view all the answers

What type of function is f(x) = 2x + 3?

<p>Linear function (B)</p> Signup and view all the answers

What operation can be used to combine two or more functions?

<p>Composition (A)</p> Signup and view all the answers

What is the purpose of finding the inverse of a function?

<p>To find the function that 'reverses' the original function (A)</p> Signup and view all the answers

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Study Notes

Equations

  • An equation is a statement that says two expressions are equal.
  • It consists of two parts: the left-hand side (LHS) and the right-hand side (RHS), separated by an equal sign (=).
  • The goal is to find the value of the variable(s) that makes the equation true.
  • Types of equations:
    • Linear equation: degree of the variable(s) is 1 (e.g., 2x + 3 = 5)
    • Quadratic equation: degree of the variable(s) is 2 (e.g., x^2 + 4x + 4 = 0)
    • Exponential equation: involves exponential functions (e.g., 2^x = 8)
  • Solving equations:
    • Addition/Subtraction property: add/subtract the same value to both sides
    • Multiplication/Division property: multiply/divide both sides by the same non-zero value
    • Distributive property: expand and simplify expressions

Functions

  • A function is a relation between a set of inputs (domain) and a set of possible outputs (range).
  • It is denoted by f(x) or g(x), where x is the input or independent variable.
  • Notation:
    • f(x) = expression: defines the function
    • f(a) = output: evaluates the function at input a
  • Types of functions:
    • Linear function: f(x) = mx + b, where m is the slope and b is the y-intercept
    • Quadratic function: f(x) = ax^2 + bx + c, where a, b, and c are constants
    • Exponential function: f(x) = a^x, where a is the base
  • Function operations:
    • Domain and range: find the set of inputs and outputs
    • Composition: combine two or more functions
    • Inverse: find the function that "reverses" the original function

Graphing

  • The graph of a function is a visual representation of the relationship between the input and output values.
  • Coordinates:
    • x-axis: horizontal axis, represents the input values
    • y-axis: vertical axis, represents the output values
  • Graph types:
    • Linear graph: straight line
    • Quadratic graph: parabola (opens upward or downward)
    • Exponential graph: curved line that increases/decreases rapidly
  • Graphing techniques:
    • Plotting points: find and plot points on the graph
    • Using intercepts: find the x-intercept (where the graph crosses the x-axis) and y-intercept (where the graph crosses the y-axis)
    • Using symmetry: identify symmetries about the x-axis, y-axis, or origin

Equations

  • An equation consists of two expressions separated by an equal sign (=) with the goal of finding the value of the variable(s) that makes the equation true.
  • There are different types of equations, including:
    • Linear equations, where the degree of the variable(s) is 1 (e.g., 2x + 3 = 5)
    • Quadratic equations, where the degree of the variable(s) is 2 (e.g., x^2 + 4x + 4 = 0)
    • Exponential equations, which involve exponential functions (e.g., 2^x = 8)
  • To solve equations, properties such as:
    • Addition/Subtraction property can be used to add/subtract the same value to both sides
    • Multiplication/Division property can be used to multiply/divide both sides by the same non-zero value
    • Distributive property can be used to expand and simplify expressions

Functions

  • A function is a relation between a set of inputs (domain) and a set of possible outputs (range) denoted by f(x) or g(x).
  • Functions can be defined using notation such as f(x) = expression, which defines the function, and f(a) = output, which evaluates the function at input a.
  • There are different types of functions, including:
    • Linear functions, where f(x) = mx + b, with m as the slope and b as the y-intercept
    • Quadratic functions, where f(x) = ax^2 + bx + c, with a, b, and c as constants
    • Exponential functions, where f(x) = a^x, with a as the base
  • Function operations include:
    • Finding the domain and range of a function
    • Composing two or more functions
    • Finding the inverse of a function

Graphing

  • The graph of a function is a visual representation of the relationship between the input and output values.
  • The graph has a:
    • x-axis, which represents the input values
    • y-axis, which represents the output values
  • Different types of graphs include:
    • Linear graphs, which are straight lines
    • Quadratic graphs, which are parabolas (opening upward or downward)
    • Exponential graphs, which are curved lines that increase/decrease rapidly
  • Graphing techniques include:
    • Plotting points to find and plot points on the graph
    • Using intercepts to find the x-intercept and y-intercept
    • Using symmetry to identify symmetries about the x-axis, y-axis, or origin

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