Basic Operations
| Site: | Clare-Gladwin RESD |
| Course: | Michigan Algebra II KHauck |
| Book: | Basic Operations |
| Printed by: | Guest user |
| Date: | Tuesday, November 26, 2024, 12:29 AM |
Description
Basic Operations
Is it a Function?
Remember from Algebra I that a relation is a function if every input has exactly one output. The following links provide an opportunity to review some of the basic concepts of functions.
To review how to identify a function from a table, select the following link:
Functions from Tables
To review how to identify a function from a graph, select the following link:
Functions from Graphs
Guided Practice
To solidify your understanding of identifying if a relation is a function, visit the following link to Holt, Rinehart and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.
Evaluating Functions
Functions are represented in math using parentheses such as f(x). This notation indicates “f†is a function of, or depends on, the variable x. A linear equation might be y = 2x – 4, while the equivalent function would be f(x) = 2x – 4. Both the equation and function would create the same table and the same graph. However, function notation states the input and the output at the same time, something the equation cannot do.
Example 1 Given f(x) = 3x – 4, find f(-2).
Step 1. Eliminate the input value “x†and replace it with parentheses.
f ( ) = 3( ) – 4
Step 2. Place the input value of -2 inside the parentheses.
f (-2) = 3(-2) – 4
Step 3. Simplify the right side of the equation.
f (-2) = -6 – 4
f (-2) = - 10
*Note: This is read, “The function f, with an input of -2, has an output of -10.
This can be graphed as the ordered pair (-2, -10).
Example 2
Given g(x) = x2 – 4, find g(3).g ( ) = ( )2 – 4
g (3) = (3)2 – 4
g (3) = 9 – 4
*Note: This is read, “The function g, with an input of 3, has an output of 5.
This can be graphed as the ordered pair (3, 5).
Adding Functions
The four main operations of addition, subtraction, multiplication, and division can also be performed with functions. During this course, these basic operations will be further discussed in the polynomial, rational, exponential, and logarithmic units.To add functions, combine like terms.
Example Given f(x) = 3x2 +4x – 5 and g(x) = 2x2 - 6x + 5, find f(x) + g(x).
f (x) + g(x)
(3x2 +4x – 5) + (2x2 - 6x + 5)
3x2 +4x – 5 + 2x2 - 6x + 5
(3x2 + 2x2) + (4x – 6x) + (-5 + 5)
5x2 – 2x
Subtracting Functions
To subtract functions, first change the sign of every term being subtracted using the Distributive Property. Eliminate the parentheses and combine like terms.Example Given f(x) = 3x2 +4x – 5 and g(x) = 2x2 - 6x + 5, find f(x) - g(x).
f(x) - g(x)
(3x2 + 4x – 5) - (2x2 - 6x + 5)
(3x2 +4x – 5) + (- 2x2 + 6x – 5)
3x2 +4x – 5 - 2x2 + 6x – 5
(3x2– 2x2) + (4x + 6x) + (-5 – 5)
x 2 + 10x – 10
Video Lesson: Add & Subtract
To learn how to add and subtract functions select the following link:Adding and Subtracting Functions
Multiplying Functions
To multiply functions, replace the equivalent expressions and then multiply using the Distributive Property. Remember, the Product Rule of Exponents states that when multiplying terms with the same base, add the exponents.
For example: xm ? xn = x(m+n).
Example Given f(x) = 3x– 5 and g(x) = 6x + 5, find f(x) · g(x).
f(x) · g(x)
(3x – 5) · (6x + 5)
(3x)(6x) + (3x)(5) + (-5)(6x) + (-5)(5)
18x2 + 15x – 30x – 25
18x2 – 15x – 25
Dividing Functions
To divide functions, put the first function as the numerator of a fraction and the second function as the denominator of a fraction. Simplify if possible. Always define your function; division by zero is not defined. Therefore, values that would create a zero in the denominator are not allowed. This concept will be addressed in more detail in the rational functions unit.
Example 1 Given
Step 1. Put the functions in fraction form.
Step 2. Try to simplify the new expression.
This expression is already simplified.
Step 3. Be sure your expression is defined.
Since division by zero is undefined, 
.
Therefore, f(x) ÷ g(x) = 
; when 
.
Example 2
Given
and 
, find f(x) ÷ g(x). 
Since division by zero is undefined, 
.
Therefore, f(x) ÷ g(x) = 
; when .
Video Lesson: Multiply & Divide
To learn how to multiply and divide functions select the following link:Multiplying and Dividing Functions
Practice
Families of Functions Worksheet*Note: If Google Docs displays “Sorry, we were unable to retrieve the document for viewing,†refresh your browser.
Answer Key
Families of Functions Answer Key*Note: If Google Docs displays “Sorry, we were unable to retrieve the document for viewing,†refresh your browser.
Sources
Sources used in this book:
Embracing Mathematics, Assessment & Technology in High Schools; a Michigan Mathematics & Science Partnership Grant Project
Holt, Rinehart & Winston, "Determine Whether a Relation is a Function." http://my.hrw.com/math06_07/nsmedia/lesson_videos/alg2/player.html?contentSrc=7045/7045.xml (accessed 8/16/2010).
Holt, Rinehart & Winston, "Foundations for Functions." http://my.hrw.com/math06_07/nsmedia/homework_help/alg2/alg2_ch01_06_homeworkhelp.html (accessed 8/16/2010).
Holt, Rinehart & Winston, "Properties and Attributes of Functions." http://my.hrw.com/math06_07/nsmedia/homework_help/alg2/alg2_ch09_04_homeworkhelp.html (accessed 6/29/2010).
Holt, Rinehart & Winston, "Using the Vertical Line Test." http://my.hrw.com/math06_07/nsmedia/lesson_videos/alg2/player.html?contentSrc=7046/7046.xml (accessed 8/16/2010).
Kenny Felder, "Function Concepts -- Introduction," Connexions, December 30, 2008, http://cnx.org/content/m18192/1.2/.
Kenny Felder, "Function Concepts -- What is a Function?," Connexions, December 30, 2008, http://cnx.org/content/m18189/1.2/.
Kenny Felder, "Function Concepts -- Function Notation," Connexions, April 15, 2009, http://cnx.org/content/m18188/1.3/.