BAAS 105 - Functions Study Notes

BAAS 105 - Section 1.2 - Functions

Definitions

  • Relation

    • A set of ordered pairs.

  • Function

    • A special type of relation where each domain (input) corresponds to exactly one element in the range (output).

    • A function can be represented in multiple forms:

    • Set of ordered pairs

    • Table

    • Equation

    • Graph

  • Domain

    • All possible x-values of a function.

  • Range

    • All possible y-values of a function.

Examples and Non-examples of Functions

  • Examples

    • Relation satisfying the function definition.

  • Non-Examples

    • Relation not satisfying the function definition (clarifications and references to specific cases can be developed).

Graphing a Function

  • Graphing the function:

    • Example: Graph y=3x2y = 3x^2.

  • Graphing another function:

    • Example: Graph y=2x+4y = 2x + 4.

Determining if the Graph of a Relation is a Function

  • The process of determining if a given graph represents a function involves applying the vertical line test:

    • If a vertical line intersects the graph at more than one point, then it is not a function.

Evaluating Functions

  • Function Notation:

    • The notation used is y=f(x)y = f(x), where the function is defined.

    • Example: For the function f(x)=3x2+2x−1f(x) = 3x^2 + 2x - 1

    • To evaluate:

      • f(3)f(3)

      • f(−2)f(-2)

      • f(a)f(a)

  • Another function example:

    • f(x)=x−4+rac92x2f(x) = x - 4 + rac{9}{2x^2}

    • Evaluations could include:

      • f(3)f(3)

      • f(−2)f(-2)

      • f(a)f(a)

Mortgage Payments Overview

  • Mortgage Payments Table:

    • The table includes the number of years to pay off a $100,000 mortgage at various interest rates when paying $800 per month.

    • Rate (r) (%) | Function f(r) (Years)

      • 2.4 | 12

      • 5.2 | 15

      • 6.3 | 17

      • 7.4 | 20

      • 8.7 | 27

  • Example Questions regarding Mortgage Payments:

    • a) What is f(6.3)f(6.3), and what does it represent?

    • b) If f(r)=20f(r) = 20, what is the corresponding value of rr?

    • c) Does the relationship f(2.4+6.3)=f(2.4)+f(2.3)f(2.4 + 6.3) = f(2.4) + f(2.3) hold?

The Difference Quotient

  • Definition:

    • The difference quotient is given by the formula f(x+h)−f(x)h\frac{f(x + h) - f(x)}{h}.

    • Example evaluation for f(x)=x2f(x) = x^2:

    • Find f(x+h)−f(x)h\frac{f(x + h) - f(x)}{h}.

    • Another evaluation example for f(x)=2x2+3x+1f(x) = 2x^2 + 3x + 1:

    • Find f(x+h)−f(x)h\frac{f(x + h) - f(x)}{h}.

Domain and Range

  • Domain:

    • Refers to all possible x-values of a function.

  • Range:

    • Refers to all possible y-values of a function.

  • Examples:

    • For the function y=4x2y = 4x^2, find the domain and range.

    • For the function y=4−xy = \sqrt{4 - x}, determine the domain and range.

    • For the function y=1+1x−2y = 1 + \frac{1}{x - 2}, find the domain and range.

Operations with Functions

  • Sum of Two Functions:

    • Defined as (f+g)(x)=f(x)+g(x)(f + g)(x) = f(x) + g(x).

    • Example: Let f(x)=3x+1f(x) = 3x + 1 and g(x)=x2−2g(x) = x^2 - 2.

  • Difference of Two Functions:

    • Defined as (f−g)(x)=f(x)−g(x)(f - g)(x) = f(x) - g(x).

    • Example: Using the same ff and gg as before.

  • Product of Two Functions:

    • Defined as (f⋅g)(x)=f(x)⋅g(x)(f \cdot g)(x) = f(x) \cdot g(x).

    • Example: Using the same functions.

  • Quotient of Two Functions:

    • Defined as (f÷g)(x)=f(x)g(x)(f \div g)(x) = \frac{f(x)}{g(x)}.

    • Example: Again using the functions from before.

  • Composite Functions:

    • The composite functions are defined as follows:

    • (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x))

    • (g∘f)(x)=g(f(x))(g \circ f)(x) = g(f(x))

    • Example with f(x)=3x+1f(x) = 3x + 1 and g(x)=x2−2g(x) = x^2 - 2.

Applications from the Book

  • Specific examples and applications discussed during the course, requiring further elaboration based on textbook references.

Homework and Practice

  • Suggested assignments:

    • Problems to work on from the textbook: 3, 5, 11, 13, 19, 21, 25, 35-42.

    • Checkpoints (green boxes in the textbook) highlighted as good practice!

Note: Additional context and specific examples requested in each section could potentially enhance comprehension and application of the function concepts discussed above.