Properties of Functions by Shin Thant Moe

What is a Function?

stmoe1 — 2023-12-11 01:40:16 UTC

A function is a relation that assigns each input exactly one output (x and y).

How to Identify Functions on a Graph?

stmoe1 — 2023-12-11 01:43:00 UTC

A vertical line test can be used to identify if the graph is a function.

How to evaluate Functions on a Formula or on a Graph?

stmoe1 — 2023-12-11 01:53:10 UTC

It can be compared to a "machine" where each input(x) follows a rule to produce its output called f(x). f(x) is read as "f of x" which equals to y.

Example:

Evaluate

f(x)=3x+9 for x=5,

substitute 5 for the variable x and write,

f(5)= 3(5)+9

f(5)=15+9

f(5)= 24

Different types of Graphs depending on the Function

stmoe1 — 2023-12-11 02:15:34 UTC

What is an Inverse Function?

stmoe1 — 2023-12-11 02:19:08 UTC

The Inverse Function is defined by exchanging the Independent and Dependent Variables of f(x). An Inverse Function will reverse the actions of a function. f ^{- 1} is read as "f inverse of x."

Steps to find the Inverse Function

stmoe1 — 2023-12-11 02:23:39 UTC

f(x) = 8x-3 1. Write the function and replace f(x) with y.

y = 8x-3 2. Switch the x and y variables.

x = 8y-3 3. Solve for y.

x+3/8 = y 4. Replace y with f(x).

f(x) = x+3/8

One-to-One Functions

stmoe1 — 2023-12-11 02:28:38 UTC

A function is one-to-one if no two x-values have the same y-values. A horizontal line test can be performed to determine if the graph is one-to-one.

What is Domain and Range?

stmoe1 — 2023-12-11 02:33:11 UTC

Domain(Input): A set of all possible values that represent the independent variable or the x-values listed as a discrete set of values or as an inequality from least to greatest.

Range(Output): A set of all possible values that represent the dependent variable or the y-values listed as a discrete set of values or as an inequality from least to greatest.

How to find Domain and Range from a Graph

stmoe1 — 2023-12-11 02:35:21 UTC

Domain: Find the domain from left to right on the x-axis.

Range: Find the range from bottom to top on the y-axis.

What are all real numbers?

stmoe1 — 2023-12-11 02:46:18 UTC

All real value can be identified for the domain/range. In this situation, we would say that the domain/range is all real numbers, sometimes written as x/y is an element of all real numbers. (xER/yER).

What does the arrow suggest?

The arrow shows that the lines continues indefinitely.

Guiding Questions

stmoe1 — 2023-12-11 02:49:52 UTC

What are the highest and lowest values that make sense in the context?
What type of numbers make sense in the context? (All Real Numbers, Integers, Whole Numbers)
Should the values be written as a discrete set or as a continuous interval?