Chapter02-Updated

Chapter 2: Functions and Graphs

Introduction to mathematical analysis for various fields including business, economics, and sciences.

Understand functions and domains.
Introduce different types of functions.
Learn operations: addition, subtraction, multiplication, division, and multiplication by a constant.
Introduce inverse functions and their properties.
Graph equations and functions.
Study symmetry about the x- and y-axis.
Familiarize with shapes of the graphs of six basic functions.

Definition: A function f from set A to set B is a correspondence that assigns each element in A to exactly one element in B.
- Notation: f: A → B
- Domain (D) is set A; Codomain (C) is set B; and Range (R) = {f(x): x ∈ A}.
Example
- A = {a,b,c,d} maps uniquely to B = {x, y, z, w}.
- A case where f is not a function because one element in B is mapped from two in A.
The domain considered is the set of real numbers or subsets thereof.

Each input corresponds to one output.
Domain: All possible input values.
Range: All possible output values.
Equality of Functions: Two functions are equal if they have the same domain and produce identical outputs for all inputs.

Examples demonstrating equal vs. unequal functions through specific domains and outputs.
Key takeaway is that even if functions output the same values, differing domains affect equality.

Functions f and g demonstrate different outputs based on their defined domains. Determination of equality based on conditions dictated by the domain.

Examples of evaluating functions and determining their respective domains.
- Function notations and sample calculations to illustrate solving for function values.
- Determining domain restrictions for polynomial and rational functions.

Specific examples discussed regarding the domains of various types of functions, including polynomial and square root functions. Calculation of domains incorporating restrictions such as non-negativity and defined denominators.

Constant Function: Defined as h(x) = c for any constant c, with domain covering all real numbers.
Polynomial Functions: Functions expressed in polynomial form have inherent degrees based on the leading coefficient.
- Classifications based on degrees: Linear (degree 1), Quadratic (degree 2), Cubic (degree 3).
Rational Functions: Quote demonstrating the nature of rational functions as combinations of polynomials.
Absolute Value Functions: Discussed relationships and outputs based on the input’s sign.

Definition and example calculations of factorial functions, illustrating their computation.

Operations of combining functions is essential, displaying methods to compute sums, differences, and products of given functions.
Composition of functions clearly defined, showcasing relationships between functions through examples.

Discussed how to identify inverse functions through equality of conditions when applying f and g iteratively.
Description of how to find an inverse function algebraically through manipulation of the original function’s equation.

Introduction to the graphical representation of functions in 2D coordinate systems.
Definition and calculation of x-intercepts and y-intercepts, with examples provided.
Exploration of specific cases for intercept calculations.

Explanation of functions involving multiple variables, illustrated by an example concerning company profits based on quantities of different souvenirs.
Further examples computing function values with specified pairs of inputs.
Emphasis on the application of functions in real-world scenarios and analytical computations.

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