Summary of What Is a Function? | Precalculus

Main Ideas and Concepts

• Definition of a Function:

  A Function is a specific type of relationship between inputs and outputs, typically represented as a relationship between two sets of numbers. Each input (from the domain) must correspond to exactly one output (from the range). If an input corresponds to multiple outputs, it is not a Function.

• Function Notation:

  Commonly represented as f(x), where f is the Function and x is the input variable. Examples include Linear Functions like f(x) = 3x + 5 and y = 2x - 7.

• Function Tables:

  To determine if a relation is a Function, create a Function table by substituting different values for x and observing the corresponding y values. Each unique x value must yield a unique y value.

• Identifying Functions from Tables:

  Analyze Function tables to determine if they represent functions:

  • Valid Function: Different x values with unique y values.
  • Not a Function: Same x value leading to different y values.
• Graphical Representation:

  Use the Vertical Line Test: If a vertical line intersects a graph at more than one point, it is not a Function.

• Types of Functions:
  • Linear Functions: Straight lines, e.g., y = mx + b.
  • Quadratic Functions: Parabolic shapes, e.g., y = x^2.
  • Cubic Functions: Graphs shaped like an "S", e.g., y = x^3.
  • Radical Functions: Functions involving roots, e.g., y = √x.
  • Logarithmic Functions: e.g., y = log(x).
  • Exponential Functions: e.g., y = e^x or y = 3^x.
  • Trigonometric Functions: Periodic functions like y = sin(x).
  • Absolute Value Functions: e.g., y = |x|.
  • Rational Functions: e.g., y = 1/x.
  • Polynomial Functions: e.g., y = x^4 - x^3 + 2x^2 - 7.
• Evaluating Functions:

  Substitute specific values into the Function to find outputs. For multivariable functions, substitute values for each variable accordingly.

Methodology for Evaluating Functions

• Single-variable Function Evaluation:

  Given f(x) = x^2 + 4x - 7, to find f(3):

  • Substitute x with 3: f(3) = 3^2 + 4(3) - 7
  • Calculate: 9 + 12 - 7 = 14.
• Multivariable Function Evaluation:

  Given f(x, y) = 2x^2 - y^2 + 3xy, to find f(2, 3):

  • Substitute x with 2 and y with 3: f(2, 3) = 2(2^2) - (3^2) + 3(2)(3)
  • Calculate: 8 - 9 + 18 = 17.

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