Representing Functions

Representing Functions

• Words: Least mathematical way to define functions.
• Table: Collection of data obtained from experiments.
• Graph: Provides the most illustrative representation.
• Formula: Most compact way to represent functions.

Representing Functions using Graphs

• Linear Functions: Straight line graph.
• Piecewise Functions: Different definitions based on domain partitions.
  • Examples include tax brackets with varying tax rates.
  • If all pieces are linear, the function is piecewise linear.
• Power and Root Functions:
  • Power Functions: y = x^n
    • Even n: Graph symmetric about the y-axis.
    • Odd n: Graph symmetric about the origin.
  • Root Functions: y= x^(1/n)
    • Even n: Graph is in the first and second quadrants.
    • Odd n: Graph is in all four quadrants.
• Rational Functions: Functions defined as the ratio of two polynomials.

Representing Functions using Formulas

• Composite Functions: A function applied to the result of another function.
  • Notation: f(g(x))
  • Domain: The domain of the composite function is limited by the inner function's range and the outer function's domain.

Graph Transformation

• Vertical Shift: y = f(x) + c
  • Shifts the graph c units upward if c is positive.
  • Shifts the graph c units downward if c is negative.
• Horizontal Shift: y = f(x - c)
  • Shifts the graph c units right if c is positive.
  • Shifts the graph c units left if c is negative.
• Vertical Scale: y = cf(x)
  • Stretches the graph vertically if c > 1.
  • Compresses the graph vertically if 0 < c < 1.
  • Reflects the graph across the x-axis if c < 0.
• Horizontal Scale: y = f(cx)
  • Compresses the graph horizontally if c > 1.
  • Stretches the graph horizontally if 0 < c < 1.
  • Reflects the graph across the y-axis if c < 0.

Representing Functions

• Functions can be represented in various ways:
  • Words: A descriptive, less mathematical approach.
  • Table: Shows data collected through experiments.
  • Graph: Offers the most visual understanding.
  • Formula: The most concise way to express many functions.

Linear Functions

• Linear functions are characterized by a constant rate of change.
• Their graph is a straight line.

Piecewise Functions

• These functions have different definitions based on specific parts of their domain.
• An example is income tax, where different tax brackets have varying rates.
• A piecewise linear function has linear segments for each part of its definition.

Power and Root Functions

• Power Functions:
  • When the exponent (n) is even, the graph is symmetrical about the y-axis.
  • When the exponent (n) is odd, the graph is symmetrical about the origin.
• Root Functions:
  • When the root (n) is even, the function is defined only for non-negative inputs.
  • When the root (n) is odd, the function is defined for all real numbers.

Formula Representation

• Formulas offer a concise way to represent functions.
• Common examples include:
  • f(x) = x^2 (quadratic function)
  • f(x) = sin(x) (sine function)
  • f(x) = ln(x) (natural logarithm function)

Composite Functions

• A composite function is created by substituting one function into another.
• The output of the inner function becomes the input of the outer function.
• Notation: f(g(x)) or f ◦ g(x)

Domain of Composite Functions

• The domain of a composite function is limited by the inner function's output, which must be included in the outer function's domain.

Graph Transformation

• Graph transformations involve altering the shape, position, or orientation of a function's graph.
• Types of transformations:
  • Vertical Shift: Moving the graph up or down by adding a constant to the function.
  • Horizontal Shift: Moving the graph left or right by subtracting/adding a constant inside the function.
  • Vertical Scale & Reflection: Scaling the graph vertically by multiplying by a constant; a negative constant reflects the graph across the x-axis.
  • Horizontal Scale & Reflection: Scaling the graph horizontally by dividing by a constant; a negative constant reflects the graph across the y-axis.

Book & Lecture Links

• Access to lecture materials and the textbook is provided through a Google Drive link.

This quiz explores various methods for representing functions including through words, tables, graphs, and formulas. It covers linear, piecewise, power, root, and rational functions along with their unique characteristics. Test your understanding of these concepts and enhance your mathematical skills.