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.