Function Operations I. Ever since basic math, you have been combining numbers by using addition, subtraction, multiplication, and division. Add: 5 + Subtract: 7 Multiply: (9)(0) Divide: (5) () or 5 II. Addition, subtraction, multiplication, and division are called Operations. III. DEFINITION: An operation is a procedure that is used to combine two items in order to make a new, third item. Sometimes the word element is used instead o the word item. 5 + = 7 New item (new element) Another item (or element) Operation One item (or element) IV. The BASIC OPERATIONS are: ADDITION MULTIPLICATION SUBTRACTION DIVISION V. Throughout your classes in math, you have worked with operations, but the items that you combined weren t always just numbers. VI. The study o operations is a mathematical thread that is ound throughout all algebra classes.
D I F F E R E N T I T E M S (E L E M E N T S) Operation Basic Math Numbers Algebra Epressions Algebra Polynomials Algebra Radicals Addition 5 + 4 0+ 7 ( + ) + (5 + + ) + Subtraction 7-8y y (4 + ) ( 7+ ) 7 9 9 Multiplication ()(7) ( y)(0 ) (+ )( + 4) ( )(5 ) Division 40 8 or (0 ) ( ) or ( ) ( + + ) or (4 ) ( ) 40 0 4 8 or + + VII. Questions or Intermediate Algebra:. I the operations work with numbers, epressions, polynomials, and radicals, then do the operations also work with unctions?. How do we deine the procedures or combining unctions using +?,,, VIII. FUNCTION OPERATIONS ADDITION ( + g)( ) Add like terms o to the like terms o g. Simpliy. Do NOT multiply by. The () means to write the answer using the letter as the variable. Likewise, (+g)() means to put where there is an in and g. Get numbers or and g, and add the results. It does NOT mean to multiply by. EXAMPLE ( ) = +, g ( ) = 4 Find: ( + g)( ) ( g)( ) + = + + = + + = 4 4 EXAMPLE =, g = + ( ) ( ) Find: ( + g)( ) and ( + g)() + g = + + = + + ( )( ) + g = + + = + + = ( )( ) ( ) ( ) 4 4.
SUBTRACTION ( g)( ) Subtract the terms in g rom the terms in. Simpliy. Here s how to do it: Write sets o empty parentheses with a negative sign between them. ( ) ( ). Put the terms o into the irst parentheses, and the terms o g into the second one. Distribute the negative sign to all o the terms in the second parentheses. Then combine like terms with. Do NOT multiply by. The () net to (-g) means to write the answer using the letter as the variable. ( ) = +, g ( ) = 4 Find: ( g)( ) = + ( g)( ) ( 4) = + = = + 4 + + 4 5+ Now Find: ( g)( ) Put - into ( g)( ) and simpliy. ( g)( ) = 5+ g = + ( )( ) ( ) 5( ) = + 5+ = + = MULTIPLICATION ( g)( ) Multiply all terms o g by all the terms in. Simpliy. Here s how: Write sets o empty parentheses net to each other. ( )( ). Put all o the terms o into the irst parentheses, and all o the terms o g into the second one. Start with the irst term o and multiply ALL terms o g by it. Then take the second term o and multiply all o the terms o g by it. Continue or each term o you ind in the irst parentheses. When you have multiplied all o the terms o g by the last term o, combine like terms. Do NOT multiply by. The () net to (g) means to write the answer using the letter as the variable.
( ) = ( + ) g ( ) = + Find: ( g)( ) ( g) ( ) = ( + ) ( + ) = + + + ( ) ( ) () ( ) ( ) () = + + + 4 = + + 8 DIVISION ( )( ) g Divide by g. Simpliy i possible. Identiy restrictions. Here s how: Write and g net to each other with a division sign between them. Net, write the reciprocal o g and change the division to multiplication. Simpliy i possible by canceling common actors (you can NOT cancel terms). To ind the restrictions, look at the denominator o and determine what would make it zero. State can not equal that value. Do the same or g and do the same when you write the reciprocal o g beore you cancel. Other restrictions are ound by looking under even radicals (square roots, ourth roots, etc. One can t obtain a real number when there is a negative under an even radical. EXAMPLE ( ) = g ( ) = 4 Find: ( )( ) g ( )( ) = note: 4 g 4 = note: 0 4 = ( 4) = 0,4 ( 4) EXAMPLE ( ) = g ( ) = 9 Find: ( )( ) g ( )( ) =, ; g 9 = 9 =,0, 9 0
COMPOSITION ( D g)( ) ( gd )( ) g is second, so put g into where is. Simpliy. is second, so put into g where is. Simpliy EXAMPLE EXAMPLE ( ) = +, g( ) = ( ) =, g ( ) = 5+ Find: a). ( D g)( ) b). ( gd )( ) ( )( ) Dg = ( g ( ) ) + ( ) = + = + 4 ( gd ) ( ) = ( ( ) ) = ( + ) = + = ( ) + Find: a). ( D g)( ) b). ( gd )( ) ( Dg)( ) = ( g ( )) = ( 5+ ) = ( 5) ( ) = 5 9 = 5 7 ( gd )( ) = 5( ( ) ) + = 5( ) + = 5( ) 5( ) + = 0 5+ = 5+ IX. DOMAIN and RANGE OF COMBINED FUNCTIONS.. Combine the unctions using the operations indicated in a given problem.. Graph the result o step. Use a graphing calculator and Y=. You may have to adjust the WINDOW by changing the min, ma, ymin, and yma.. Domain: walk along the -ais. When you see the curve above or below you, you are in the domain. 4. Range: move a horizontal yard stick up the y-ais. I the yard stick crosses the graph, you are in the range.
EXAMPLE: ( ) = + g ( ) = ( Dg)( ) = ( g ( )) + = + + + Domain: Walk along -ais. Look above and below. For what values o do you see the graph? Answer or all real values o. Domain: All Real Numbers, \ Interval notation: (, ) y = + + GRAPH: - Y = ( abs( + )) + min: -0 ymin: -0 ma: 0 yma: 0 scl: yscl: Range: move the yard stick up until it touches the graph. It irst touches at y = and continues to touch to ininity. Range: y or in interval notation [, )