Summary of "What Is a Function? | Precalculus"

Main Ideas and Concepts

Definition of a Function:
A Function is a specific type of relationship between inputs and outputs, typically represented as a relationship between two sets of numbers. Each input (from the domain) must correspond to exactly one output (from the range). If an input corresponds to multiple outputs, it is not a Function.
Function Notation:
Commonly represented as f(x), where f is the Function and x is the input variable. Examples include Linear Functions like f(x) = 3x + 5 and y = 2x - 7.
Function Tables:
To determine if a relation is a Function, create a Function table by substituting different values for x and observing the corresponding y values. Each unique x value must yield a unique y value.
Identifying Functions from Tables:
Analyze Function tables to determine if they represent functions:
- Valid Function: Different x values with unique y values.
- Not a Function: Same x value leading to different y values.
Graphical Representation:
Use the Vertical Line Test: If a vertical line intersects a graph at more than one point, it is not a Function.
Types of Functions:
- Linear Functions: Straight lines, e.g., y = mx + b.
- Quadratic Functions: Parabolic shapes, e.g., y = x^2.
- Cubic Functions: Graphs shaped like an "S", e.g., y = x^3.
- Radical Functions: Functions involving roots, e.g., y = √x.
- Logarithmic Functions: e.g., y = log(x).
- Exponential Functions: e.g., y = e^x or y = 3^x.
- Trigonometric Functions: Periodic functions like y = sin(x).
- Absolute Value Functions: e.g., y = |x|.
- Rational Functions: e.g., y = 1/x.
- Polynomial Functions: e.g., y = x^4 - x^3 + 2x^2 - 7.
Evaluating Functions:
Substitute specific values into the Function to find outputs. For multivariable functions, substitute values for each variable accordingly.

Methodology for Evaluating Functions

Single-variable Function Evaluation:
Given f(x) = x^2 + 4x - 7, to find f(3):
- Substitute x with 3: f(3) = 3^2 + 4(3) - 7
- Calculate: 9 + 12 - 7 = 14.
Multivariable Function Evaluation:
Given f(x, y) = 2x^2 - y^2 + 3xy, to find f(2, 3):
- Substitute x with 2 and y with 3: f(2, 3) = 2(2^2) - (3^2) + 3(2)(3)
- Calculate: 8 - 9 + 18 = 17.