Composition of Functions

Composition of functions is a fundamental concept in mathematics, especially in the study of functions and their properties. When you compose two functions, you are essentially applying one function to the result of another function. The composition of two functions f and g is denoted as (f o g)(x), which is read as "f composed with g of x". This is equivalent to f(g(x)).

Composition Notation

The composition of functions f and g is written as (f o g)(x) = f(g(x)). This means that you first apply the function g to the input x, and then apply the function f to the result of g(x).

Example

Let's consider two functions f(x) = 2x + 1 and g(x) = x^2. To find the composition (f o g)(x), we first evaluate g(x) and then substitute the result into f(x).

g(x) = x^2

f(g(x)) = f(x^2) = 2x^2 + 1

Properties of Composition

Some important properties of composition of functions include:

Associativity: (f o g) o h = f o (g o h)
Identity function: f o I = f, where I is the identity function

Study Guide

When studying composition of functions, it's important to:

Understand the notation and how to compute the composition of two functions
Practice evaluating compositions of functions using various examples
Learn the properties of composition, such as associativity and the identity function
Apply composition of functions to real-world problems and scenarios

By mastering the concept of composition of functions, you will develop a deeper understanding of how functions interact and how to manipulate them to solve mathematical problems.

◂Math Worksheets and Study Guides Eighth Grade. Displaying data

Organizing & Displaying Data

The resources above cover the following skills:

Data Analysis and Probability (NCTM)

Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.

Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.

Select and use appropriate statistical methods to analyze data.

Discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem-and-leaf plots, box plots, and scatterplots.

Develop and evaluate inferences and predictions that are based on data.

Make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit.

Grade 8 Curriculum Focal Points (NCTM)

Data Analysis and Number and Operations and Algebra: Analyzing and summarizing data sets

Students use descriptive statistics, including mean, median, and range, to summarize and compare data sets, and they organize and display data to pose and answer questions. They compare the information provided by the mean and the median and investigate the different effects that changes in data values have on these measures of center. They understand that a measure of center alone does not thoroughly describe a data set because very different data sets can share the same measure of center. Students select the mean or the median as the appropriate measure of center for a given purpose.

Connections to the Grade 8 Focal Points (NCTM)

Data Analysis: Building on their work in previous grades to organize and display data to pose and answer questions, students now see numerical data as an aggregate, which they can often summarize with one or several numbers. In addition to the median, students determine the 25th and 75th percentiles (1st and 3rd quartiles) to obtain information about the spread of data. They may use box-and-whisker plots to convey this information. Students make scatterplots to display bivariate data, and they informally estimate lines of best fit to make and test conjectures.