Functions and Function Notation Overview

Chapter Overview - Focus on Functions and Lines

• Sections:

  • Functions and Function Notation

  • Domain and Range

  • Rates of Change and Behavior of Graphs

  • Linear Functions

  • Graphs of Linear Functions

  • Modeling with Linear Functions

  • Fitting Linear Models to Data

Functions

• Definition: A function relates an independent variable (input) to a dependent variable (output) such that each input corresponds to exactly one output.

• Notational example: If h is a function of a, then h = f(a).

Function Relationships

• Examples of functions:

  • Height as a function of age: Clear one-to-one relationship.

  • Price as a function of item: Each item maps to one price.

  • Mathematical example: If f(x) = x^2, each input x gives exactly one output x^2. For instance, f(2) = 4 and f(-2) = 4. This is a function.

  • Percentage to decimal grade: Not always a function due to grading schemes (e.g., multiple percentages might round to the same decimal grade if not precisely defined).

• Non-example: If the relationship is x = y^2, then for x = 4, y could be 2 or -2. Since one input (4) maps to multiple outputs (2 and -2), this is not a function.

Function Notation

• Usage to express relationships clearly:

  • Example: Days in month as d = f(m) where m is the month name.

  • Evaluate by substituting into the function and obtaining output.

Representing Functions

• Functions can be represented through:

  • Words: Clear descriptions.

  • Tables: Pairs of inputs and outputs to illustrate relationships.

  • Graphs: Visual representation with x (input) and y (output).

  • Formulas: Mathematical expressions directly relating input to output.

Evaluating and Solving Functions

• Evaluating: Finding output for a given input (e.g., g(3)).

• Solving: Finding inputs that give a specific output (e.g., g(n) = 6 can yield multiple n values).

Vertical Line Test

• A graphical method to determine if a curve represents a function.

• If any vertical line intersects the graph more than once, it is not a function.

Example Evaluations and Solutions

• Evaluations use direct substitution of values into functions or tables.

  • Example Evaluation: Given the function f(x) = 2x + 1, to evaluate f(3):

  • Substitute x=3 into the function: f(3) = 2(3) + 1 = 6 + 1 = 7.

• Solutions require reversing the function to find input values based on the desired output.

  • Example Solution: Given the function f(x) = 2x + 1, to solve for x when f(x) = 9:

  • Set the function equal to the desired output: 9 = 2x + 1.

  • Solve for x:

    • Subtract 1 from both sides: 8 = 2x.

    • Divide by 2: x = 4.

Important Topics

• Functional definition clarity, notation, various representation forms, and evaluation/solution methodologies.