Functions in Algebra

Domain and Range

• Domain: The set of input values (x) for which a function is defined.
  • Expressed as {x | x is an element of the domain}
  • May be all real numbers, or a subset of real numbers
• Range: The set of output values (y) that a function can produce.
  • Expressed as {y | y is an element of the range}
  • May be all real numbers, or a subset of real numbers

Function Notation

• Function notation: A way to express a function using variables and parentheses.
  • f(x) is read as "f of x"
  • f(x) = y means "f assigns to x the value y"
  • Example: f(x) = 2x + 1

Graphing Functions

• Graphing a function: Plotting the points (x, y) that satisfy the function.
  • The graph of a function is a visual representation of the relationship between x and y.
  • The x-axis represents the domain, and the y-axis represents the range.
  • Graphing helps identify key features, such as maxima, minima, and asymptotes.

Linear Functions

• Linear function: A function with a constant rate of change.
  • General form: f(x) = mx + b, where m is the slope and b is the y-intercept.
  • Characteristics:
    • The graph is a straight line.
    • The slope (m) represents the rate of change.
    • The y-intercept (b) is the point where the line crosses the y-axis.

Constructing Functions

• Constructing a function: Building a function to model a real-world situation or satisfy certain conditions.
  • Steps:
    1. Identify the problem or situation.
    2. Determine the type of function needed (e.g., linear, quadratic, etc.).
    3. Choose a function that satisfies the conditions.
  • Example: Construct a function that models the cost of producing x units of a product, where the cost is $5 per unit plus a fixed overhead of $100.

Domain and Range

• Domain: the set of input values (x) for which a function is defined
• Domain notation: {x | x is an element of the domain}
• Range: the set of output values (y) that a function can produce
• Range notation: {y | y is an element of the range}

Function Notation

• Function notation: a way to express a function using variables and parentheses
• f(x) is read as "f of x"
• f(x) = y means "f assigns to x the value y"
• Example: f(x) = 2x + 1

Graphing Functions

• Graphing a function: plotting the points (x, y) that satisfy the function
• The graph of a function is a visual representation of the relationship between x and y
• The x-axis represents the domain, and the y-axis represents the range
• Graphing helps identify key features, such as maxima, minima, and asymptotes

Linear Functions

• Linear function: a function with a constant rate of change
• General form: f(x) = mx + b, where m is the slope and b is the y-intercept
• Characteristics:
  • The graph is a straight line
  • The slope (m) represents the rate of change
  • The y-intercept (b) is the point where the line crosses the y-axis

Constructing Functions

• Constructing a function: building a function to model a real-world situation or satisfy certain conditions
• Steps:
  • Identify the problem or situation
  • Determine the type of function needed (e.g., linear, quadratic, etc.)
  • Choose a function that satisfies the conditions
• Example: modeling the cost of producing x units of a product, where the cost is $5 per unit plus a fixed overhead of $100